Chapter 5
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1 2 3 4 5 6 7 8 9 10
Advanced Mathematical Concepts
Chapter 5 Resource Masters
XXX
11 10 09 08 07 06 05 04
Contents
Vocabulary Builder . . . . . . . . . . . . . . . . . vii-x
Lesson 5-7
Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 199
Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 201
Lesson 5-1
Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 181
Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 183
Lesson 5-8
Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 202
Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 204
Lesson 5-2
Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 184
Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 186
Chapter 5 Assessment
Chapter 5 Test, Form 1A . . . . . . . . . . . . 205-206
Chapter 5 Test, Form 1B . . . . . . . . . . . . 207-208
Chapter 5 Test, Form 1C . . . . . . . . . . . . 209-210
Chapter 5 Test, Form 2A . . . . . . . . . . . . 211-212
Chapter 5 Test, Form 2B . . . . . . . . . . . . 213-214
Chapter 5 Test, Form 2C . . . . . . . . . . . . 215-216
Chapter 5 Extended Response
Assessment . . . . . . . . . . . . . . . . . . . . . . . 217
Chapter 5 Mid-Chapter Test . . . . . . . . . . . . . 218
Chapter 5 Quizzes A & B . . . . . . . . . . . . . . . 219
Chapter 5 Quizzes C & D. . . . . . . . . . . . . . . 220
Chapter 5 SAT and ACT Practice . . . . . 221-222
Chapter 5 Cumulative Review . . . . . . . . . . . 223
Lesson 5-3
Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 187
Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 189
Lesson 5-4
Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 190
Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 192
Lesson 5-5
Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 193
Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 195
SAT and ACT Practice Answer Sheet,
10 Questions . . . . . . . . . . . . . . . . . . . . . . . A1
SAT and ACT Practice Answer Sheet,
20 Questions . . . . . . . . . . . . . . . . . . . . . . . A2
ANSWERS . . . . . . . . . . . . . . . . . . . . . . A3-A17
Lesson 5-6
Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 196
Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 198
© Glencoe/McGraw-Hill
iii
Advanced Mathematical Concepts
A Teacher’s Guide to Using the
Chapter 5 Resource Masters
The Fast File Chapter Resource system allows you to conveniently file the
resources you use most often. The Chapter 5 Resource Masters include the core
materials needed for Chapter 5. These materials include worksheets, extensions,
and assessment options. The answers for these pages appear at the back of this
booklet.
All of the materials found in this booklet are included for viewing and printing in
the Advanced Mathematical Concepts TeacherWorks CD-ROM.
Vocabulary Builder Pages vii-x include a
Practice There is one master for each lesson.
student study tool that presents the key
vocabulary terms from the chapter. Students are
to record definitions and/or examples for each
term. You may suggest that students highlight or
star the terms with which they are not familiar.
These problems more closely follow the
structure of the Practice section of the Student
Edition exercises. These exercises are of
average difficulty.
When to Use These provide additional
practice options or may be used as homework
for second day teaching of the lesson.
When to Use Give these pages to students
before beginning Lesson 5-1. Remind them to
add definitions and examples as they complete
each lesson.
Enrichment There is one master for each
lesson. These activities may extend the concepts
in the lesson, offer a historical or multicultural
look at the concepts, or widen students’
perspectives on the mathematics they are
learning. These are not written exclusively
for honors students, but are accessible for use
with all levels of students.
Study Guide There is one Study Guide
master for each lesson.
When to Use Use these masters as
reteaching activities for students who need
additional reinforcement. These pages can also
be used in conjunction with the Student Edition
as an instructional tool for those students who
have been absent.
© Glencoe/McGraw-Hill
When to Use These may be used as extra
credit, short-term projects, or as activities for
days when class periods are shortened.
iv
Advanced Mathematical Concepts
Assessment Options
Intermediate Assessment
The assessment section of the Chapter 5
Resources Masters offers a wide range of
assessment tools for intermediate and final
assessment. The following lists describe each
assessment master and its intended use.
Chapter Tests
•
Forms 1A, 1B, and 1C Form 1 tests contain
multiple-choice questions. Form 1A is
intended for use with honors-level students,
Form 1B is intended for use with averagelevel students, and Form 1C is intended for
use with basic-level students. These tests
are similar in format to offer comparable
testing situations.
Forms 2A, 2B, and 2C Form 2 tests are
composed of free-response questions. Form
2A is intended for use with honors-level
students, Form 2B is intended for use with
average-level students, and Form 2C is
intended for use with basic-level students.
These tests are similar in format to offer
comparable testing situations.
The Extended Response Assessment
includes performance assessment tasks that
are suitable for all students. A scoring
rubric is included for evaluation guidelines.
Sample answers are provided
for assessment.
© Glencoe/McGraw-Hill
•
Four free-response quizzes are included to
offer assessment at appropriate intervals in
the chapter.
•
The SAT and ACT Practice offers
continuing review of concepts in various
formats, which may appear on standardized
tests that they may encounter. This practice
includes multiple-choice, quantitativecomparison, and grid-in questions. Bubblein and grid-in answer sections are provided
on the master.
•
The Cumulative Review provides students
an opportunity to reinforce and retain skills
as they proceed through their study of
advanced mathematics. It can also be used
as a test. The master includes free-response
questions.
Answers
All of the above tests include a challenging
Bonus question.
•
A Mid-Chapter Test provides an option to
assess the first half of the chapter. It is
composed of free-response questions.
Continuing Assessment
Chapter Assessments
•
•
v
•
Page A1 is an answer sheet for the SAT and
ACT Practice questions that appear in the
Student Edition on page 341. Page A2 is an
answer sheet for the SAT and ACT Practice
master. These improve students’ familiarity
with the answer formats they may
encounter in test taking.
•
The answers for the lesson-by-lesson
masters are provided as reduced pages with
answers appearing in red.
•
Full-size answer keys are provided for the
assessment options in this booklet.
Advanced Mathematical Concepts
Chapter 5 Leveled Worksheets
Glencoe’s leveled worksheets are helpful for meeting the needs of every
student in a variety of ways. These worksheets, many of which are found
in the FAST FILE Chapter Resource Masters, are shown in the chart
below.
•
Study Guide masters provide worked-out examples as well as practice
problems.
•
Each chapter’s Vocabulary Builder master provides students the
opportunity to write out key concepts and definitions in their own
words.
•
Practice masters provide average-level problems for students who
are moving at a regular pace.
•
Enrichment masters offer students the opportunity to extend their
learning.
Five Different Options to Meet the Needs of
Every Student in a Variety of Ways
primarily skills
primarily concepts
primarily applications
BASIC
AVERAGE
1
Study Guide
2
Vocabulary Builder
3
Parent and Student Study Guide (online)
© Glencoe/McGraw-Hill
4
Practice
5
Enrichment
vi
ADVANCED
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
5
Reading to Learn Mathematics
Vocabulary Builder
This is an alphabetical list of the key vocabulary terms you will learn in Chapter 5.
As you study the chapter, complete each term’s definition or description.
Remember to add the page number where you found the term.
Vocabulary Term
Found
on Page
Definition/Description/Example
ambiguous case
angle of depression
angle of elevation
apothem
arccosine relation
arcsine relation
arctangent relation
circular function
cofunctions
cosecant
(continued on the next page)
© Glencoe/McGraw-Hill
vii
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
5
Reading to Learn Mathematics
Vocabulary Builder (continued)
Vocabulary Term
Found
on Page
Definition/Description/Example
cosine
cotangent
coterminal angles
degree
Hero’s Formula
hypotenuse
initial side
inverse
Law of Cosines
Law of Sines
leg
(continued on the next page)
© Glencoe/McGraw-Hill
viii
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
3
5
Reading to Learn Mathematics
Vocabulary Builder (continued)
Vocabulary Term
Found
on Page
Definition/Description/Example
minute
quadrant angle
reference angle
secant
second
side adjacent
side opposite
sine
solve a triangle
standard position
tangent
(continued on the next page)
© Glencoe/McGraw-Hill
ix
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
5
Reading to Learn Mathematics
Vocabulary Builder (continued)
Vocabulary Term
Found
on Page
Definition/Description/Example
terminal side
trigonometric function
trigonometric ratio
unit circle
vertex
© Glencoe/McGraw-Hill
x
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
5-1
Study Guide
Angles and Degree Measure
Decimal degree measures can be expressed in degrees(°),
minutes(′), and seconds(″).
Example 1
a. Change 12.520° to degrees, minutes, and
seconds.
12.520° 12° (0.520 60)′
Multiply the decimal portion of
12° 31.2′
the degrees by 60 to find minutes.
12° 31′ (0.2 60)″ Multiply the decimal portion of
12° 31′ 12″
the minutes by 60 to find seconds.
12.520° can be written as 12° 31′ 12″.
b. Write 24° 15′ 33″ as a decimal rounded to the
nearest thousandth.
1
° 33″ 1
°
24° 15′ 33″ 24° 15′ 60′
3600″
24.259°
24° 15′ 33″ can be written as 24.259°.
An angle may be generated by the rotation of one ray
multiple times about the origin.
Example 2
Give the angle measure represented by each
rotation.
a. 2.3 rotations clockwise
2.3 360 828
Clockwise rotations have negative measures.
The angle measure of 2.3 clockwise rotations is 828°.
b. 4.2 rotations counterclockwise
4.2 360 1512
Counterclockwise rotations have positive
measures.
The angle measure of 4.2 counterclockwise rotations is 1512°.
If is a nonquadrantal angle in standard position, its reference
angle is defined as the acute angle formed by the terminal side of
the given angle and the x-axis.
Reference
Angle Rule
Example 3
For any angle , 0° < < 360°, its reference angle ′ is defined by
a. , when the terminal side is in Quadrant I,
b. 180° , when the terminal side is in Quadrant II,
c. 180°, when the terminal side is in Quadrant III, and
d. 360° , when the terminal side is in Quadrant IV.
Find the measure of the reference angle for 220°.
Because 220° is between 180° and 270°, the
terminal side of the angle is in Quadrant III.
220° 180° 40°
The reference angle is 40°.
© Glencoe/McGraw-Hill
181
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
5-1
Practice
Angles and Degree Measure
Change each measure to degrees, minutes, and seconds.
1. 28.955
2. 57.327
Write each measure as a decimal degree to the nearest
thousandth.
3. 32 28′ 10″
4. 73 14′ 35″
Give the angle measure represented by each rotation.
5. 1.5 rotations clockwise
6. 2.6 rotations counterclockwise
Identify all angles that are coterminal with each angle. Then find
one positive angle and one negative angle that are coterminal
with each angle.
7. 43
8. 30
If each angle is in standard position, determine a coterminal angle
that is between 0 and 360, and state the quadrant in which the
terminal side lies.
9. 472
10. 995
Find the measure of the reference angle for each angle.
11. 227
12. 640
13. Navigation For an upcoming trip, Jackie plans to sail from
Santa Barbara Island, located at 33 28′ 32″ N, 119 2′ 7″ W, to
Santa Catalina Island, located at 33.386 N, 118.430 W. Write
the latitude and longitude for Santa Barbara Island as decimals
to the nearest thousandth and the latitude and longitude for
Santa Catalina Island as degrees, minutes, and seconds.
© Glencoe/McGraw-Hill
182
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
5-1
Enrichment
Reading Mathematics: If and Only If Statements
If p and q are interchanged in a conditional statement so that p
becomes the conclusion and q becomes the hypothesis, the new statement, q → p, is called the converse of p → q.
If p → q is true, q → p may be either true or false.
Example
Find the converse of each conditional.
a. p → q: All squares are rectangles. (true)
q → p: All rectangles are squares. (false)
b. p → q: If a function ƒ(x) is increasing on an
interval I, then for every a and b contained in
I, ƒ(a) ƒ(b) whenever a b. (true)
q → p: If for every a and b contained in an interval I,
ƒ(a) ƒ(b) whenever a b then function ƒ(x) is
increasing on I. (true)
In Lesson 3-5, you saw that the two statements in Example 2 can be
combined in a single statement using the words “if and only if.”
A function ƒ(x) is increasing on an interval I if and only if for
every a and b contained in I, ƒ(a) ƒ(b) whenever a b.
The statement “p if and only if q” means that p implies q and
q implies p.
State the converse of each conditional. Then tell if the converse is
true or false. If it is true, combine the statement and its converse
into a single statement using the words “if and only if.”
1. All integers are rational numbers.
2. If for all x in the domain of a function ƒ(x), ƒ(x) ƒ(x), then
the graph of ƒ(x) is symmetric with respect to the origin.
3. If ƒ(x) and ƒ1(x) are inverse functions, then [ ƒ ° ƒ1](x) [ ƒ1 ° ƒ](x) x.
© Glencoe/McGraw-Hill
183
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
5-2
Study Guide
Trigonometric Ratios in Right Triangles
The ratios of the sides of right triangles can be used to define
the trigonometric ratios known as the sine, cosine, and
tangent.
Example 1
Find the values of the sine, cosine, and tangent
for A.
.
First find the length of BC
2
2
2
(AC) (BC) (AB)
Pythagorean Theorem
2
2
2
10 (BC) 20
Substitute 10 for AC and 20 for AB.
2
(BC) 300
BC 3
0
0
or 103
Take the square root of each side.
Disregard the negative root.
Then write each trigonometric ratio.
side opposite
sin A hypotenuse
side adjacent
cos A hypotenuse
side opposite
tan A side adjacent
10
3 or 3
0 or 1
10
3 or 3
sin A cos A 12
tan A 20
2
0
10
2
Trigonometric ratios are often simplified but never written as
mixed numbers.
Three other trigonometric ratios, called cosecant, secant,
and cotangent, are reciprocals of sine, cosine, and tangent,
respectively.
Example 2
Find the values of the six trigonometric
ratios for R.
First determine the length of the hypotenuse.
Pythagorean Theorem
(RT)2 (ST)2 (RS)2
2
2
2
RT 15, ST 3
15 3 (RS)
2
(RS) 234
RS 2
3
4
or 32
6
Disregard the negative root.
side opposite
side adjacent
side opposite
sin R hypotenuse
cos R hypotenuse
6
2
sin R 3 or 26
32
6
5
2
6
tan R 3 or 1
cos R 15 or 26
15
5
3
2
6
hypotenuse
hypotenuse
tan R side adjacent
side adjacent
csc R side opposite
sec R side adjacent
cot R side opposite
3
6
2
or 2
csc R 6
3
3
2
6
or 6
2
sec R 15
5
5 or 5
cot R 13
© Glencoe/McGraw-Hill
184
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
5-2
Practice
Trigonometric Ratios in Right Triangles
Find the values of the sine, cosine, and tangent for each B.
1.
2.
3. If tan 5, find cot .
4. If sin 38, find csc .
Find the values of the six trigonometric ratios for each S.
5.
6.
7. Physics Suppose you are traveling in a car when a beam of light
passes from the air to the windshield. The measure of the angle
of incidence is 55, and the measure of the angle of refraction is
sin i
n, to find the index of refraction n
35 15′. Use Snell’s Law, sin r
of the windshield to the nearest thousandth.
© Glencoe/McGraw-Hill
185
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
5-2
Enrichment
Using Right Triangles to Find the Area of
Another Triangle
You can find the area of a right triangle by using the formula
A
1
bh.
2
In the formula, one leg of the right triangle can be used as
the base, and the other leg can be used as the height.
The vertices of a triangle can be represented on the coordinate plane
by three ordered pairs. In order to find the area of a general triangle,
you can encase the triangle in a rectangle as shown in the diagram
below.
A rectangle is placed around the triangle so that the vertices of the
triangle all touch the sides of the rectangle.
Example
Find the area of a triangle whose vertices are
A(1, 3), B(4, 8), and C(8, 5).
Plot the points and draw the triangle. Encase the triangle
in a rectangle whose sides are parallel to the
axes, then find the coordinates of the vertices of the
rectangle.
Area ABC area ADEF area ADB area BEC area CFA, where ADB, BEC,
and CFA are all right triangles.
Area ABC 5(9) 1
(5)(5)
2
1
(4)(3)
2
1
(2)(9)
2
17.5 square units
Find the area of the triangle having vertices with each set of coordinates.
1. A(4, 6), B(–1, 2), C(6, –5)
2. A(–2, –4), B(4, 7), C(6, –1)
3. A(4, 2), B(6, 9), C(–1, 4)
4. A(2, –3), B(6, –8), C(3, 5)
© Glencoe/McGraw-Hill
186
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
5-3
Study Guide
Trigonometric Functions on the Unit Circle
Example 1
Use the unit circle to find cot (270°).
The terminal side of a 270° angle in
standard position is the positive y-axis,
which intersects the unit circle at (0, 1).
By definition, cot (270°) xy or 01.
Therefore, cot (270°) 0.
Trigonometric
Functions of
an Angle in
Standard Position
y
sin r
r
csc y
cos x
tan y
x
x
x
cot y
r
r
sec Example 2 Find the values of the six trigonometric functions
for angle in standard position if a point with
coordinates (9, 12) lies on its terminal side.
We know that x 9 and y 12. We need to
find r.
Example 3
y2
r x2
Pythagorean Theorem
r (
9
)2
2
12
Substitute 9 for x and 12 for y.
r 2
2
5
or 15
Disregard the negative root.
sin 1125 or 45
9 or 3
cos 15
5
12
or 4
tan 3
9
csc 1152 or 54
15
or 5
sec 3
9
9 or 3
cot 12
4
Suppose is an angle in standard position whose
terminal side lies in Quadrant I. If cos 35, find
the values of the remaining five trigonometric
functions of .
r2 x2 y2
52 32 y2
16 y2
4y
Pythagorean Theorem
Substitute 5 for r and 3 for x.
Take the square root of each side.
Since the terminal side of lies in Quadrant I, y must
be positive.
sin 45
csc 54
© Glencoe/McGraw-Hill
tan 43
sec 35
187
cot 34
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
5-3
Practice
Trigonometric Functions on the Unit Circle
Use the unit circle to find each value.
1. csc 90
2. tan 270
3. sin (90)
Use the unit circle to find the values of the six trigonometric
functions for each angle.
4. 45
5. 120
Find the values of the six trigonometric functions for angle in
standard position if a point with the given coordinates lies on its
terminal side.
6. (1, 5)
7. (7, 0)
8. (3, 4)
© Glencoe/McGraw-Hill
188
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
5-3
Enrichment
Areas of Polygons and Circles
A regular polygon has sides of equal length and angles of equal
measure. A regular polygon can be inscribed in or circumscribed
about a circle. For n-sided regular polygons, the following area
formulas can be used.
AC r2
Area of circle
nr2
2
sin
360°
n
Area of circumscribed polygon AC nr2 tan
180°
n
Area of inscribed polygon
AI Use a calculator to complete the chart below for a unit circle (a circle of radius 1).
Number
of Sides
Area of
Inscribed
Polygon
Area of Circle
less
Area of Polygon
Area of
Circumscribed
Polygon
Area of Polygon
less
Area of Circle
3
1.2990381
1.8425545
5.1961524
2.0545598
1.
4
2.
8
3.
12
4.
20
5.
24
6.
28
7.
32
8.
1000
9. What number do the areas of the circumscribed and inscribed
polygons seem to be approaching as the number of sides of the
polygon increases?
© Glencoe/McGraw-Hill
189
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
5-4
Study Guide
Applying Trigonometric Functions
Trigonometric functions can be used to solve problems
involving right triangles.
Example 1
If T 45° and u 20, find t to the nearest tenth.
From the figure, we know the measures of an
angle and the hypotenuse. We want to know the
measure of the side opposite the given angle. The
sine function relates the side opposite the angle
and the hypotenuse.
sin T ut
sin 45° 2t0
20 sin 45° t
14.14213562 t
side opposite
sin hypotenuse
Substitute 45° for T and 20 for u.
Multiply each side by 20.
Use a calculator.
Therefore, t is about 14.1.
Example 2
Geometry The apothem of a regular polygon is the
measure of a line segment from the center of the
polygon to the midpoint of one of its sides. The
apothem of a regular hexagon is 2.6 centimeters.
Find the radius of the circle circumscribed about
the hexagon to the nearest tenth.
First draw a diagram. Let a be the angle
measure formed by a radius and its adjacent
apothem. The measure of a is 360° 12 or
30°. Now we know the measures of an angle
and the side adjacent to the angle.
2.6
cos 30° r
r cos 30° 2.6
side adjacent
cos hypotenuse
Multiply each side by r.
2.
6
Divide each side by cos 30°.
r cos 30°
r 3.0022214
Use a calculator.
Therefore, the radius is about 3.0 centimeters.
© Glencoe/McGraw-Hill
190
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
5-4
Practice
Applying Trigonometric Functions
Solve each problem. Round to the nearest tenth.
1. If A 55 55′ and c 16, find a.
2. If a 9 and B 49, find b.
3. If B 56 48′ and c 63.1, find b.
4. If B 64 and b 19.2, find a.
5. If b 14 and A 16, find c.
6. Construction A 30-foot ladder leaning against
the side of a house makes a 70 5′ angle with the
ground.
a. How far up the side of the house does the
ladder reach?
b. What is the horizontal distance between the
bottom of the ladder and the house?
7. Geometry A circle is circumscribed about a regular hexagon with an
apothem of 4.8 centimeters.
a. Find the radius of the circumscribed circle.
b. What is the length of a side of the hexagon?
c. What is the perimeter of the hexagon?
8. Observation A person standing 100 feet from the bottom
of a cliff notices a tower on top of the cliff. The angle of
elevation to the top of the cliff is 30. The angle of elevation
to the top of the tower is 58. How tall is the tower?
© Glencoe/McGraw-Hill
191
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
5-4
Enrichment
Making and Using a Hypsometer
A hypsometer is a device that can be used to measure the height of
an object. To construct your own hypsometer, you will need a
rectangular piece of heavy cardboard that is at least 7 cm by 10 cm,
a straw, transparent tape, a string about 20 cm long, and a small
weight that can be attached to the string.
Mark off 1-cm increments along one short side and one long side of
the cardboard. Tape the straw to the other short side. Then attach
the weight to one end of the string, and attach the other end of the
string to one corner of the cardboard, as shown in the figure below.
The diagram below shows how your hypsometer should look.
To use the hypsometer, you will need to measure the distance from
the base of the object whose height you are finding to where you
stand when you use the hypsometer.
Sight the top of the object through the straw. Note where the freehanging string crosses the bottom scale. Then use similar triangles
to find the height of the object.
1. Draw a diagram to illustrate how you can use similar triangles
and the hypsometer to find the height of a tall object.
Use your hypsometer to find the height of each of the following.
2. your school’s flagpole
3. a tree on your school’s property
4. the highest point on the front wall of your school building
5. the goal posts on a football field
6. the hoop on a basketball court
7. the top of the highest window of your school building
8. the top of a school bus
9. the top of a set of bleachers at your school
10. the top of a utility pole near your school
© Glencoe/McGraw-Hill
192
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
5-5
Study Guide
Solving Right Triangles
When we know a trigonometric value of an angle but not the
value of the angle, we need to use the inverse of the
trigonometric function.
Trigonometric Function
y sin x
y cos x
y tan x
Example 1
Inverse Trigonometric Relation
x sin1 y or x arcsin y
x cos1 y or x arccos y
x tan1 y or x arctan y
.
Solve tan x 3
If tan x 3
, then x is an angle whose tangent
.
is 3
x arctan 3
From a table of values, you can determine that
x equals 60°, 240°, or any angle coterminal with
these angles.
Example 2
If c 22 and b 12, find B.
In this problem, we know the side opposite the
angle and the hypotenuse. The sine function
relates the side opposite the angle and the
hypotenuse.
side opposite
sin B bc
sin hypotenuse
sin B 1222
Substitute 12 for b and 22 for c.
B sin11222
Definition of inverse
B 33.05573115 or about 33.1°.
Example 3
Solve the triangle where b 20 and c 35, given
the triangle above.
cos A bc
2
0
cos A 35
a2 b2 c2
a2 202 352
a 8
2
5
a 28.72281323
20
A cos-1
35 A 55.15009542
55.15009542 B 90
B 34.84990458
Therefore, a 28.7, A 55.2°, and B 34.8°.
© Glencoe/McGraw-Hill
193
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
5-5
Practice
Solving Right Triangles
Solve each equation if 0 x 360.
2
1. cos x 2
2. tan x 1
1
3. sin x 2
Evaluate each expression. Assume that all angles are in Quadrant I.
3
4. tan tan1 3
5. tan cos1 23
6. cos arcsin 15
3
Solve each problem. Round to the nearest tenth.
7. If q 10 and s 3, find S.
8. If r 12 and s 4, find R.
9. If q 20 and r 15, find S.
Solve each triangle described, given the triangle at the right.
Round to the nearest tenth, if necessary.
10. a 9, B 49
11. A 16, c 14
12. a 2, b 7
13. Recreation The swimming pool at Perris Hill Plunge is 50 feet
long and 25 feet wide. The bottom of the pool is slanted so that
the water depth is 3 feet at the shallow end and 15 feet at the
deep end. What is the angle of elevation at the bottom of the
pool?
© Glencoe/McGraw-Hill
194
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
5-5
Enrichment
Disproving Angle Trisection
Most geometry texts state that it is impossible to trisect an arbitrary
angle using only a compass and straightedge. This fact has been
known since ancient times, but since it is usually stated without
proof, some geometry students do not believe it. If the students set
out to find a method for trisecting angles, they will probably try the
following method. It is based on the familiar construction which
allows a segment to be divided into any desired number of congruent
segments. You can use inverse trigonometric functions to show that
application of the method to the trisection of angles is not valid.
Given: A
Claim: A can be trisected using the following method.
Method: Choose point C on one ray of A .
Through C construct a perpendicular to
the other ray, intersecting it at B.
Construct M and N, the points that
into three congruent
divide CB
segments. Draw A
M
and AN
, which
trisect CAB into the congruent angles
1, 2, and 3.
The proposed method has been used to construct the figure below.
CM MN NB 1. AB 5. Follow the instructions to show that
the three angles 1, 2, and 3, are not congruent. Find angle
measures to the nearest tenth of a degree.
1. Express m3 as the value of an inverse function.
2. Find the measure of 3.
3. Write mMAB as the value of an inverse function.
4. Find the measure of MAB.
5. Find the measure of 2.
6. Find mCAB and use it to find m 1.
7. Explain why the proposed method for trisecting an angle fails.
© Glencoe/McGraw-Hill
195
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
5-6
Study Guide
The Law of Sines
Given the measures of two angles and one side of a triangle,
we can use the Law of Sines to find one unique solution for
the triangle.
Law of Sines
Example 1
a b c
sin A
sin B
sin C
Solve ABC if A 30°, B 100°, and a 15.
First find the measure of C.
C 180° (30° 100°) or 50°
Use the Law of Sines to find b and c.
a b
sin A
sinB
15
b
sin 30°
sin 100°
15
s
in
1
0
0°
b
sin 30°
c a
sinC
sin A
c
15
sin 50°
sin 30°
15 sin
50°
c
sin 30°
29.54423259 b
c 22.98133329
Therefore, C 50°, b 29.5, and c 23.0.
The area of any triangle can be expressed in terms of two
sides of a triangle and the measure of the included angle.
Area (K) of a Triangle
Example 2
K 12 bc sin A
K 12 ac sin B K 12 ab sin C
Find the area of ABC if a 6.8, b 9.3, and C 57°.
K 12ab sin C
K 12(6.8)(9.3) sin 57°
K 26.51876336
The area of ABC is about 26.5 square units.
© Glencoe/McGraw-Hill
196
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
5-6
Practice
The Law of Sines
Solve each triangle. Round to the nearest tenth.
1. A 38, B 63, c 15
2. A 33, B 29, b 41
3. A 150, C 20, a 200
4. A 30, B 45, a 10
Find the area of each triangle. Round to the nearest tenth.
5. c 4, A 37, B 69
6. C 85, a 2, B 19
7. A 50, b 12, c 14
8. b 14, C 110, B 25
9. b 15, c 20, A 115
10. a 68, c 110, B 42.5
11. Street Lighting A lamppost tilts toward the sun
at a 2 angle from the vertical and casts a 25-foot
shadow. The angle from the tip of the shadow to the
top of the lamppost is 45. Find the length of the
lamppost.
© Glencoe/McGraw-Hill
197
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
5-6
Enrichment
Triangle Challenge
A surveyor took the following measurements from two irregularlyshaped pieces of land. Some of the lengths and angle measurements
are missing. Find all missing lengths and angle measurements. Round
lengths to the nearest tenth and angle measurements to the nearest
minute.
e
1.
G
a
H
2.
c
f
b
e
J
© Glencoe/McGraw-Hill
198
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
5-7
Study Guide
The Ambiguous Case for the Law of Sines
Case 1: A 90° for a, b, and A
a b sin A
no solution
a b sin A
one solution
a b
one solution
b sin A a b two solutions
Case 2: A 90°
a b
no solution
ab
one solution
If we know the measures of two sides and a
nonincluded angle of a triangle, three situations
are possible: no triangle exists, exactly one
triangle exists, or two triangles exist. A triangle
with two solutions is called the ambiguous case.
Example
Find all solutions for the triangle
if a 20, b 30, and A 40°. If no
solutions exist, write none.
Since 40° 90°, consider Case 1.
b sin A 30 sin 40°
b sin A 19.28362829
Since 19.3 20 30, there are two solutions for
the triangle.
Use the Law of Sines to find B.
20
sin 40°
30
sin B
a b
sin A
sin B
30 sin
40°
sin B 20
30 sin
40°
B sin1
20
B 74.61856831
So, B 74.6°. Since we know there are two solutions,
there must be another possible measurement for B.
In the second case, B must be less than 180° and
have the same sine value. Since we know that if
90, sin sin (180 ), 180° 74.6° or 105.4°
is another possible measure for B. Now solve the
triangle for each possible measure of B.
Solution I
Solution II
C 180° (40° 74.6°) or 65.4°
C 180° (40° 105.4°) or 34.6°
a c
sin A
sin C
20
c
sin 40°
sin 65.4°
20 sin
65.4°
c
sin 40°
a c
sin A
sin C
20
c
sin 40°
sin 34.6°
20 sin
34.6°
c
sin 40°
c 17.66816088
c 28.29040558
One solution is B 74.6°,
C 65.4°, and c 28.3.
© Glencoe/McGraw-Hill
Another solution is B 105.4°,
C 34.6°, and c 17.7.
199
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
5-7
Practice
The Ambiguous Case for the Law of Sines
Determine the number of possible solutions for each triangle.
1. A 42, a 22, b 12
2. a 15, b 25, A 85
3. A 58, a 4.5, b 5
4. A 110, a 4, c 4
Find all solutions for each triangle. If no solutions exist, write
none. Round to the nearest tenth.
5. b 50, a 33, A 132
6. a 125, A 25, b 150
7. a 32, c 20, A 112
9. A 42, a 22, b 12
8. a 12, b 15, A 55
10. b 15, c 13, C 50
11. Property Maintenance The McDougalls plan to fence a triangular parcel of their
land. One side of the property is 75 feet in length. It forms a 38 angle with another
side of the property, which has not yet been measured. The remaining side of the
property is 95 feet in length. Approximate to the nearest tenth the length of fence
needed to enclose this parcel of the McDougalls’ lot.
© Glencoe/McGraw-Hill
200
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
5-7
Enrichment
Spherical Triangles
Spherical trigonometry is an extension of plane trigonometry.
Figures are drawn on the surface of a sphere. Arcs of great
circles correspond to line segments in the plane. The arcs of
three great circles intersecting on a sphere form a spherical
triangle. Angles have the same measure as the tangent lines
drawn to each great circle at the vertex. Since the sides are
arcs, they too can be measured in degrees.
The sum of the sides of a spherical triangle is less than 360°.
The sum of the angles is greater than 180° and less than 540°.
The Law of Sines for spherical triangles is as follows.
sin c
sin a
sin b
sin A
sin B
sin C
There is also a Law of Cosines for spherical triangles.
cos a cos b cos c sin b sin c cos A
cos b cos a cos c sin a sin c cos B
cos c cos a cos b sin a sin b cos C
Example
Solve the spherical triangle given a 72°, b 105°, and c 61°.
Use the Law of Cosines.
0.3090 (–0.2588)(0.4848) (0.9659)(0.8746) cos A
cos A 0.5143
A 59°
–0.2588 (0.3090)(0.4848) (0.9511)(0.8746) cos B
cos B –0.4912
B 119°
0.4848 (0.3090)(–0.2588) (0.9511)(0.9659) cos C
cos C 0.6148
C 52°
Check by using the Law of Sines.
sin 72°
sin 59°
sin 105°
sin 119°
sin 61°
sin 52°
1.1
Solve each spherical triangle.
1. a 56°, b 53°, c 94°
2. a 110°, b 33°, c 97°
3. a 76°, b 110°, C 49°
4. b 94°, c 55°, A 48°
© Glencoe/McGraw-Hill
201
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
5-8
Study Guide
The Law of Cosines
When we know the measures of two sides of a triangle and
the included angle, we can use the Law of Cosines to find
the measure of the third side. Often times we will use both
the Law of Cosines and the Law of Sines to solve a triangle.
Law of Cosines
Example 1
a2 b2 c2 2bc cos A
b2 a2 c2 2ac cos B
c2 a2 b2 2ab cos C
Solve ABC if B 40°, a 12, and c 6.
b2 a2 c2 2ac cos B
b2 122 62 2(12)(6) cos 40°
b2 69.68960019
b 8.348029719
So, b 8.3.
b sin B
8.
3 sin 40°
sin C c
Law
sin C
6
sin C
6 sin
40°
8.3
6 sin
40°
sin1 8.3
C
C 27.68859159
Law of Cosines
of Sines
So, C 27.7°.
A 180° (40° 27.7°) 112.3°
The solution of this triangle is b 8.3, A 112.3°, and C 27.7°.
Example 2
Find the area of ABC if a 5, b 8, and c 10.
First, find the semiperimeter of ABC.
s 12(a b c)
s 12(5 8 10)
s 11.5
Now, apply Hero’s Formula
s
)(
as
)(
bs
)
c
k s(
k 1
1
.5
(1
1
.5
)(
51
1
.5
)(
81
1
.5
0
1)
k 3
9
2
.4
3
7
5
k 19.81003534
The area of the triangle is about 19.8 square units.
© Glencoe/McGraw-Hill
202
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
5-8
Practice
The Law of Cosines
Solve each triangle. Round to the nearest tenth.
1. a 20, b 12, c 28
2. a 10, c 8, B 100
3. c 49, b 40, A 53
4. a 5, b 7, c 10
Find the area of each triangle. Round to the nearest tenth.
5. a 5, b 12, c 13
6. a 11, b 13, c 16
7. a 14, b 9, c 8
8. a 8, b 7, c 3
9. The sides of a triangle measure 13.4 centimeters,
18.7 centimeters, and 26.5 centimeters. Find the measure
of the angle with the least measure.
10. Orienteering During an orienteering hike, two hikers
start at point A and head in a direction 30 west of south
to point B. They hike 6 miles from point A to point B. From
point B, they hike to point C and then from point C back to
point A, which is 8 miles directly north of point C. How
many miles did they hike from point B to point C?
© Glencoe/McGraw-Hill
203
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
5-8
Enrichment
The Law of Cosines and the Pythagorean Theorem
The law of cosines bears strong similarities to the
Pythagorean Theorem. According to the Law of
Cosines, if two sides of a triangle have lengths a and
b and if the angle between them has a measure of x,
then the length, y, of the third side of the triangle
can be found by using the equation
y2 a2 b2 2ab cos (x °).
Answer the following questions to clarify the relationship between
the Law of Cosines and the Pythagorean Theorem.
1. If the value of x becomes less and less, what number is cos (x °)
close to?
2. If the value of x is very close to zero but then increases, what
happens to cos (x °) as x approaches 90?
3. If x equals 90, what is the value of cos (x°)? What does the equation of y2 a2 b2 2ab cos (x °) simplify to if x equals 90?
4. What happens to the value of cos (x°) as x increases beyond 90
and approaches 180?
5. Consider some particular values of a and b, say 7 for a and
19 for b. Use a graphing calculator to graph the equation you
get by solving y2 72 192 2(7)(19) cos (x °) for y.
a. In view of the geometry of the situation, what range of values
should you use for X on a graphing calculator?
b. Display the graph and use the TRACE function. What do the
maximum and minimum values appear to be for the function?
c. How do the answers for part b relate to the lengths 7 and 19?
Are the maximum and minimum values from part b ever
actually attained in the geometric situation?
© Glencoe/McGraw-Hill
204
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
5
Chapter 5 Test, Form 1A
Write the letter for the correct answer in the blank at the right of
each problem.
1. Change 128.433° to degrees, minutes, and seconds.
A. 128° 25′ 58″ B. 128° 25′ 59″ C. 128° 25′ 92″ D. 128° 26′ 00″
2. Write 43° 18′ 35″ as a decimal to the nearest thousandth of a degree.
A. 43.306°
B. 43.308°
C. 43.309°
D. 43.310°
3. Give the angle measure represented by 3.25 rotations clockwise.
A. 1170°
B. 90°
C. 90°
D. 1170°
4. Identify all coterminal angles between 360° and 360°
for the angle 420°.
A. 60° and 300°
B. 30° and 330°
C. 30° and 330°
D. 60° and 300°
5. Find the measure of the reference angle for 1046°.
A. 56°
B. 56°
C. 34°
D. 34°
6. Find the value of the tangent for A.
2
5
5
B. A. 2
2
C. 23
4. ________
5. ________
6. ________
1
4
D. 3
9. If cot θ 0.85, find tan θ .
A. 0.588
B. 0.85
C. 1.176
10. Find cos (270°).
A. undefined
B. 1
C. 1
11. Find the exact value of sec 300°.
2
3
C. 2
A. 2
B. 3
8. ________
D.
1
sec θ
9. ________
D. 1.7
10. ________
D. 0
11. ________
2
3
D. 3
12. Find the value of csc θ for angle θ in standard position if
the point at (5, 2) lies on its terminal side.
2
9
2
2
9
2
9
5
29
B. C. D. A. 2
29
5
29
12. ________
13. Suppose θ is an angle in standard position whose terminal side
2 , find the value of sec θ .
lies in Quadrant II. If sin θ 11
3
© Glencoe/McGraw-Hill
3. ________
7. ________
8. Which of the following is equal to csc θ ?
1
1
1
A. B. C. sin θ
cos θ
tan θ
A. 153
2. ________
5
D. 3
7. Find the value of the secant for R.
7
0
3
14
A. B. 5
14
5
C. 3
1. ________
3
B. 15
C. 152
205
13. ________
13
D. 12
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
5
Chapter 5 Test, Form 1A (continued)
For Exercises 14 and 15, refer to the figure. The angle of
elevation from the end of the shadow to the top of the
building is 63° and the distance is 220 feet.
14. Find the height of the building to the nearest foot.
A. 100 ft
B. 196 ft
C. 432 ft
D. 112 ft
14. ________
15. Find the length of the shadow to the nearest foot.
A. 100 ft
B. 196 ft
C. 432 ft
D. 112 ft
15. ________
16. If 0° ≤ x ≤ 360°, solve the equation sec x 2.
A. 150° and 210°
B. 210° and 330°
C. 120° and 240°
D. 240° and 300°
16. ________
17. Assuming an angle in Quadrant I, evaluate csc cot1 43 .
A. 35
B. 53
C. 45
D. 54
17. ________
18. Given the triangle at the right, find B to the
nearest tenth of a degree if b 10 and c 14.
A. 44.4°
B. 35.5°
C. 54.5°
D. 45.6°
18. ________
For Exercises 19 and 20, round answers to the nearest tenth.
19. In ABC, A 27° 35′, B 78° 23′, and c 19. Find a.
A. 8.6
B. 9.2
C. 12.8
D. 19.4
19. ________
20. If A 42.2°, B 13.6°, and a 41.3, find the area of ABC.
A. 138.8 units2 B. 493.8 units2 C. 327.4 units2 D. 246.9 units2
20. ________
21. Determine the number of possible solutions if A 62°, a 4,
and b 6.
A. none
B. one
C. two
D. three
21. ________
22. Determine the greatest possible value for B if A 30°, a 5,
and b 8.
A. 23.1°
B. 53.1°
C. 126.9°
D. 96.9°
22. ________
For Exercises 23-25, round answers to the nearest tenth.
23. In ABC, A 47°, b 12, and c 8. Find a.
A. 6.3
B. 8.7
C. 8.8
D. 18.4
23. ________
24. In ABC, a 7.8, b 4.2, and c 3.9. Find B.
A. 15.1°
B. 148.7°
C. 78.9°
24. ________
D. 16.2°
25. If a 22, b 14, and c 30, find the area of ABC.
B. 121.0 units2 C. 130.2 units2 D. 143.8 units2
A. 33 units2
25. ________
Bonus The terminal side of an angle θ in standard position
Bonus: ________
coincides with the line 4x y 0 in Quadrant II. Find sec θ
to the nearest thousandth.
A. 0.243
B. 4.123
C. 0.243
D. 4.123
© Glencoe/McGraw-Hill
206
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
5
Chapter 5 Test, Form 1B
Write the letter for the correct answer in the blank at the right of
each problem.
1. Change 110.23° to degrees, minutes, and seconds.
A. 110° 13′ 00″ B. 110° 13′ 8″ C. 110° 13′ 28″ D. 110° 13′ 48″
2. Write 24° 38′ 42″ as a decimal to the nearest thousandth of a degree.
A. 24.645°
B. 24.646°
C. 24.647°
D. 24.648°
1. ________
2. ________
3. Give the angle measure represented by 1.75 rotations counterclockwise. 3. ________
A. 630°
B. 90°
C. 90°
D. 630°
4. Identify the coterminal angle between 360° and 360° for the angle 120°. 4. ________
A. 240°
B. 60°
C. 240°
D. 300°
5. Find the measure of the reference angle for 295°.
A. 25°
B. 65°
C. 25°
D. 65°
5. ________
6. Find the value of the cosine for A.
6. ________
A. 12
3
B. 2
3
C. 3
D. 2
7. Find the value of the cosecant for R.
6
A. 2
6
B. 3
1
5
C. 3
1
0
D. 2
8. Which of the following is equal to sec ?
1
1
1
A. B. C. sin cos tan 9. If tan 0.25, find cot .
A. 0.25
B. 4
10. Find cot (180°).
A. undefined
B. 1
7. ________
8. ________
D.
1
sec 9. ________
C. 0.5
D. 14
C. 1
D. 0
10. ________
11. Find the exact value of tan 240°.
3
B. C. 3
A. 3
3
11. ________
D.
3
3
12. Find the value of sec for angle in standard position if the
point at (2, 4) lies on its terminal side.
5
5
B. 5
C. D. 5
A. 2
2
12. ________
13. Suppose is an angle in standard position whose terminal side
lies in Quadrant III. If sin 1123, find the value of cot .
13. ________
A. 15
3
© Glencoe/McGraw-Hill
3
B. 15
C. 15
2
207
3
D. 11
2
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
5
Chapter 5 Test, Form 1B (continued)
For Exercises 14 and 15, refer to the figure. The angle
of elevation from the end of the shadow to the top of
the building is 56° and the distance is 120 feet.
14. Find the height of the building to the nearest foot.
A. 99 ft
B. 67 ft
C. 178 ft
D. 81 ft
14. ________
15. Find the length of the shadow to the nearest foot.
A. 99 ft
B. 67 ft
C. 178 ft
D. 81 ft
15. ________
16. If 0° x 360°, solve the equation tan x 1.
16. ________
A. 135° and 315° B. 45° and 225° C. 45° and 315° D. 225° and 315°
17. Assuming an angle in Quadrant I, evaluate tan cos1 45.
A. 34
B. 53
C. 45
17. ________
D. 54
18. Given the triangle at the right, find A to the nearest
tenth of a degree if b 10 and c 14.
A. 44.4°
B. 35.5°
C. 54.5°
D. 45.6°
18. ________
For Exercises 19 and 20, round answers to the nearest tenth.
19. In ABC, A 41° 15′, B 107° 39′, and c 19. Find b.
A. 10.0
B. 24.3
C. 35.1
D. 54.6
19. ________
20. If A 52.6°, B 49.8°, and a 33.8, find the area of ABC.
A. 117.9 units2 B. 338.2 units2 C. 536.4 units2 D. 1072.8 units2
20. ________
21. Determine the number of possible solutions if A 62°, a 7, and b 6.
A. none
B. one
C. two
D. three
21. ________
For Exercises 22–25, round answers to the nearest tenth.
22. Determine the least possible value for B if A 30°, a 5,
and b 8.
A. 23.1°
B. 53.1°
C. 126.9°
D. 96.9°
23. In ABC, B 52°, a 14, and c 9. Find b.
A. 8.2
B. 11.0
C. 11.1
24. In ABC, a 7.8, b 4.2, and c 3.9. Find A.
A. 15.1°
B. 78.9°
C. 148.7°
22. ________
23. ________
D. 18.4
24. ________
D. 16.2°
25. If a 32, b 26, and c 40, find the area of ABC.
B. 121.0 units2 C. 298.6 units2 D. 415.2 units2
A. 49 units2
Bonus The terminal side of an angle in standard position
coincides with the line 2x y 0 in Quadrant III.
Find cos to the nearest ten-thousandth.
A. 0.4472
B. 0.4472
C. 0.8944
D. 0.8944
© Glencoe/McGraw-Hill
208
25. ________
Bonus: ________
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
5
Chapter 5 Test, Form 1C
Write the letter for the correct answer in the blank at the right of
each problem.
1. Change 36.3 to degrees, minutes, and seconds.
A. 36 18′ 00″ B. 36 18′ 16″ C. 36 18′ 24″ D. 36 18′ 28″
1. ________
2. Write 21 44′ 3″ as a decimal to the nearest thousandth of a degree.
A. 21.731
B. 21.732
C. 21.733
D. 21.734
2. ________
3. Give the angle measure represented by 0.5 rotation clockwise.
A. 180
B. 90
C. 90
D. 180
3. ________
4. Identify the coterminal angle between 0 and 360 for the angle 480.
A. 30
B. 60
C. 120
D. 240
4. ________
5. Find the measure of the reference angle for 235.
A. 125
B. 55
C. 25
D. 55
5. ________
6. Find the value of the sine for A.
1
9
1
1
19
B. A. 12
5
6. ________
C. 152
2
D. 15
7. Find the value of the cotangent for R.
A. 43
B. 34
C. 45
D. 54
8. Which of the following is equal to cot ?
1
1
1
A. B. C. sin cos sec 9. If cos 0.5, find sec .
A. 0.25
B. 0.5
10. Find tan 180.
A. undefined
7. ________
8. ________
D.
1
tan 9. ________
C. 1
D. 2
10. ________
B. 1
C. 1
11. Find the exact value of cos 135.
2
A. 1
B. C. 1
2
D. 0
11. ________
2
D. 2
12. Find the value of csc for angle in standard position if the point at
(3, 1) lies on its terminal side.
1
0
1
0
0
B. C. D. 3
A. 1
10
3
12. ________
13. Suppose is an angle in standard position whose terminal side lies
, find the value of tan .
in Quadrant II. If cos 112
3
1
3
2
5
13
A. 1
B. 5
C. 15
D. 2
12
13. ________
© Glencoe/McGraw-Hill
209
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
5
Chapter 5 Test, Form 1C (continued)
For Exercises 14 and 15, refer to the figure. The angle of elevation
from the end of the shadow to the top of the building is 70 and
the distance is 180 feet.
14. Find the height of the building to the nearest foot.
A. 62 ft
B. 66 ft
C. 169 ft
D. 495 ft
14. ________
15. Find the length of the shadow to the nearest foot.
A. 62 ft
B. 66 ft
C. 169 ft
D. 495 ft
15. ________
.
3
16. If 0 x 360, solve the equation sin x 2
A. 120 and 240
B. 240 and 300
C. 210 and 330
D. 150 and 210
16. ________
17. Assuming an angle in Quadrant I, evaluate cos tan1 43.
A. 35
B. 53
C. 45
17. ________
D. 54
18. Given the triangle at the right, find B to the
nearest tenth of a degree if b 8 and c 12.
A. 33.7
B. 41.8
C. 48.2
D. 56.3
18. ________
For Exercises 19 and 20, round answers to the nearest tenth.
19. In ABC, A 102 12, B 23 21, and c 19.8. Find a.
A. 8.0
B. 23.8
C. 48.8
D. 64.4
19. ________
20. If A 32.2, b 21.5, and c 11.3, find the area of ABC.
A. 129.5 units2 B. 102.8 units2 C. 64.7 units2 D. 32.6 units2
20. ________
21. Determine the number of possible solutions if A 48, a 5, and b 6.
A. none
B. one
C. two
D. three
21. ________
22. Determine the least possible value for B if A 20,
a 7, and b 11.
A. 12.6
B. 32.5
C. 147.5
D. 96.9
22. ________
For Exercises 23–25, round answers to the nearest tenth.
23. In ABC, A 52, b 9, and c 14. Find a.
A. 6.3
B. 8.7
C. 8.8
D. 11.0
23. ________
24. In ABC, a 2.4, b 8.2, and c 10.1. Find B.
A. 15.1
B. 21.7
C. 28.9
D. 33.3
24. ________
25. If a 12, b 30, and c 22, find the area of ABC.
A. 33.7 units2 B. 93.4 units2 C. 113.1 units2 D. 143.8 units2
25. ________
Bonus The terminal side of an angle in standard position
coincides with the line y 12 x in Quadrant I. Find
sin to the nearest thousandth.
A. 0.233
B. 0.447
C. 0.508
D. 0.693
© Glencoe/McGraw-Hill
210
Bonus: ________
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
5
Chapter 5 Test, Form 2A
1. Change 225.639 to degrees, minutes, and seconds.
1. __________________
2. Write 23 16′ 25″ as a decimal to the nearest thousandth of a degree. 2. ____________________
3. State the angle measure represented by 2.4 rotations clockwise.
3. ____________________
4. Identify all coterminal angles between 360 and 360 for the
angle 540.
4. ____________________
5. Find the measure of the reference angle for 562.
5. ____________________
For Exercises 6–8, refer to the figure.
6. Find the value of the sine for A.
6. ____________________
7. Find the value of the cotangent for A.
7. ____________________
8. Find the value of the secant for A.
8. ____________________
Exercises 6–8
9. If csc 2, find sin .
9. ____________________
10. Find sin (270).
10. ____________________
11. Find the exact value of cot 330.
11. ____________________
12. Find the exact value of sec for angle in standard position if
the point at (3, 2) lies on its terminal side.
12. ____________________
13. Suppose is an angle in standard position whose terminal side
12
, find the value of csc .
lies in Quadrant IV. If cos 13. ____________________
13
© Glencoe/McGraw-Hill
211
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
5
Chapter 5 Test, Form 2A (continued)
For Exercises 14 and 15, refer to the figure. The angle of elevation
from the far side of the pool to the top of the waterfall is 75, and
the distance is 185 feet.
14. Find the height of the waterfall
14. __________________
to the nearest foot.
15. Find the width across the pool 15.
to the nearest foot.
16. If 0
x
.
360, solve cot x 3
16. __________________
17. Assuming an angle in Quadrant I, evaluate sec tan1 34.
17. __________________
18. Given triangle at the right,
find B to the nearest tenth
of a degree if a 8 and b 20.
18. __________________
For Exercises 19 and 20, round answers to the nearest tenth.
19. In ABC, A 47 15′, B 58 33′, and c 23. Find a.
19. __________________
20. If A 37.2, B 17.9, and a 22.3, find the area of ABC.
20. __________________
21. Determine the number of possible solutions if A 47,
a 4, and b 5.
21. __________________
22. Determine the least possible value for c if A 30,
a 5, and b 8.
22. __________________
For Exercises 23-25, round answers to the nearest tenth.
23. In ABC, A 118, b 8, and c 6. Find a.
23. __________________
24. In ABC, a 9, b 5, and c 12. Find B.
24. __________________
25. If a 12, b 24, and c 30, find the area of ABC.
25. __________________
Bonus The terminal side of an angle in standard
Bonus: __________________
position coincides with the line 3x y 0 in
Quadrant II. Find csc to the nearest thousandth.
© Glencoe/McGraw-Hill
212
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
5
Chapter 5 Test, Form 2B
1. Change 124.63° to degrees, minutes, and seconds.
1. __________________
2. Write 48° 32′ 15″ as a decimal to the nearest thousandth of
a degree.
2. __________________
3. State the angle measure represented by 1.25 rotations
clockwise.
3. __________________
4. Identify all coterminal angles between 360° and 360° for
the angle 630°.
4. __________________
5. Find the measure of the reference angle for 310°.
5. __________________
For Exercises 6–8, refer to the figure.
6. Find the value of the cosine for A.
6. __________________
7. Find the value of the cosecant for A.
7. __________________
8. Find the value of the cotangent for A.
8. __________________
Exercises 6–8
9. If sec 4, find cos .
9. __________________
10. Find tan (180°).
10. __________________
11. Find the exact value of sec 240°.
11. __________________
12. Find the exact value of sec for angle in standard
position if the point at (4, 5) lies on its terminal side.
12. __________________
13. Suppose is an angle in standard position whose terminal 13. __________________
side lies in Quadrant IV. If cos 1123, find the value of cot .
© Glencoe/McGraw-Hill
213
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
5
Chapter 5 Test, Form 2B (continued)
For Exercises 14 and 15, refer to the figure. The angle of elevation
from the far side of the pool to the top of the waterfall is 54°, and
the distance is 230 feet.
14. Find the height of the waterfall
14. __________________
to the nearest foot.
15. Find the width across the pool
to the nearest foot.
16. If 0°
x
15. __________________
360°, solve the equation csc x 2.
16. __________________
2 .
17. Assuming an angle in Quadrant I, evaluate cos cot1 15
17. __________________
18. Given the triangle at the right,
find B to the nearest tenth of a
degree if a 12 and c 22.
18. __________________
For Exercises 19 and 20, round answers to the nearest tenth.
19. In ABC, A 42°, B 68°, and c 15. Find a.
19. __________________
20. If A 27.2°, B 67.4°, and a 12.8, find the area of ABC.
20. __________________
21. Determine the number of possible solutions if A 110°,
a 5, and b 4.
21. __________________
For Exercises 22–25, round answers to the nearest tenth.
22. Determine the greatest possible value for c if A 30°,
a 5, and b 8.
22. __________________
23. In ABC, A 59°, b 12, and c 4. Find a.
23. __________________
24. In ABC, a 4, b 11, and c 8. Find B.
24. __________________
25. If a 21, b 15, and c 28, find the area of ABC.
25. __________________
Bonus The terminal side of an angle in standard
Bonus:
position coincides with the line 3x y 0 in
Quadrant III. Find sin to the nearest ten-thousandth.
© Glencoe/McGraw-Hill
214
_________________
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
5
Chapter 5 Test, Form 2C
1. Change 25.6 to degrees, minutes, and seconds.
1. __________________
2. Write 75 30′ as a decimal to the nearest thousandth of a
degree.
2. __________________
3. State the angle measure represented by 1.5 rotations
counterclockwise.
3. __________________
4. Identify a coterminal angle between 0 and 360 for the
angle 225.
4. __________________
5. Find the measure of the reference angle for 235.
5. __________________
For Excercises 6–8, refer to the figure.
6. Find the value of the sine for A.
6. __________________
7. Find the value of the cotangent for A.
7. __________________
8. Find the value of the secant for A.
8. __________________
Exercises 6–8
9. If tan 3, find cot .
9. __________________
10. Find tan 180.
10. __________________
11. Find the exact value of cos 330.
11. __________________
12. Find the exact value of sin for angle in standard position 12. __________________
if the point at (1, 4) lies on its terminal side.
13. Suppose is an angle in standard position whose
terminal side lies in Quadrant II. If sin 1123, find
the value of sec .
© Glencoe/McGraw-Hill
215
13. __________________
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
5
Chapter 5 Test, Form 2C (continued)
For Exercises 14 and 15, refer to the figure. The angle of elevation
from the far side of the pool to the top of the waterfall is 68 and
the distance is 200 feet.
14. Find the height of the waterfall
14. __________________
to the nearest foot.
15. Find the width across the pool 15.
to the nearest foot.
16. If 0
x
.
3
360, solve sin x 2
16. __________________
2 .
17. Assuming an angle in Quadrant I, evaluate cos tan1 15
18. Given the triangle at the right,
find B to the nearest tenth
of a degree if b 12 and c 18.
17. __________________
18. __________________
For Exercises 19 and 20, round answers to the nearest tenth.
19. In ABC, A 47, B 58, and b 23. Find a.
19. __________________
20. If C 37.2, a 17.9, and b 22.3, find the area of ABC. 20. __________________
21. Determine the number of possible solutions if A 47,
a 2, and b 4.
21. __________________
22. Determine the greatest possible value for c if A 15,
a 8, and b 13.
22. __________________
For Exercises 23–25, round answers to the nearest tenth.
23. In ABC, A 67, b 10, and c 5. Find a.
23. __________________
24. In ABC, a 8, b 6, and c 12. Find C.
24. __________________
25. If a 18, b 22, and c 30, find the area of ABC.
25. __________________
Bonus The terminal side of an angle in standard
Bonus: __________________
position coincides with the line y x in
Quadrant I. Find tan to the nearest thousandth.
© Glencoe/McGraw-Hill
216
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
5
Chapter 5 Open-Ended Assessment
Instructions: Demonstrate your knowledge by giving a clear,
concise solution to each problem. Be sure to include all
relevant drawings and justify your answers. You may show
your solution in more than one way or investigate beyond the
requirements of the problem.
) lies on the terminal side of an angle in
1. The point at (3, 3
standard position.
a. Give the degree measure of three angles that fit the description.
b. Tell how to find the cosine of such angles. Give the cosine of these angles.
c. Name angles in the first, second, and fourth quadrants that have the same
reference angle as those above.
d. Write the coordinates of a point in Quadrant II. Find the values
of the six trigonometric functions of an angle in standard position
with this point on its terminal side.
2. A children’s play area is being built next to a circular fountain in the
park. A fence will be erected around the play area for safety. A diagram
of the area is shown below.
a. How long will the fence need to be in order to enclose the area?
b. The park commission is planning to enlarge the play area. Do you
think it should be enlarged to the east or to the west? Why?
© Glencoe/McGraw-Hill
217
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
5
Chapter 5 Mid-Chapter Test (Lessons 5-1 through 5-4)
1. Change 65.782 to degrees, minutes, and seconds.
1. __________________
2. If a 470 angle is in standard position, determine a
coterminal angle that is between 0 and 360. State the
quadrant in which the terminal side lies.
2. __________________
For Exercises 3 and 4, use right triangle ABC to find each value.
3. Find the value of the cosine for A.
3. __________________
4. Find the value of the cotangent for A.
4. __________________
5. If csc 3, find sin .
5. __________________
6. Use the unit circle to find tan 180.
6. __________________
7. Find the exact value of sin 300.
7. __________________
8. Find the value of csc for angle in standard position if
the point at (2, 1) lies on its terminal side.
8. __________________
For Exercises 9 and 10, use right triangle ABC to find each
value to the nearest tenth.
9. Find b.
9. __________________
10. Find c.
10. __________________
© Glencoe/McGraw-Hill
218
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
5
Chapter 5, Quiz A (Lessons 5-1 and 5-2)
1. Change 47.283 to degrees, minutes, and seconds.
2. Write 122 43′ 12″ as a decimal to the nearest thousandth
of a degree.
3. Give the angle measure represented by 2.25 rotations
counterclockwise.
4. Identify all coterminal angles between 360 and 360
for the angle 480.
5. Find the measure of the reference angle for 323.
6. Find the value of the sine for A.
7. Find the value of the tangent for A.
8. Find the value of the secant for A.
1. __________________
2. __________________
4. __________________
9. If cot 23, find tan .
9. __________________
3. __________________
5.
6.
7.
8.
Exercises 6–8
10. If sin 0.5, find csc .
__________________
__________________
__________________
__________________
10. __________________
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
5
Chapter 5, Quiz B (Lessons 5-3 and 5-4)
1. __________________
Use the unit circle to find each value.
1.
3.
4.
5.
sin (90)
2. csc 180
Find the exact value of cos 210.
Find the exact value of tan 135.
Find the value of sec for angle in standard position if
the point at (4, 5) lies on its terminal side.
6. Suppose is an angle in standard position whose terminal
, find the value of sin .
side lies in Quadrant III. If tan 15
2
Refer to the figure. Find each value
to the nearest tenth.
7. Find a.
8. Find c.
2.
3.
4.
5.
__________________
__________________
__________________
__________________
6. __________________
7. __________________
8. __________________
A 100-foot cable is stretched from a stake in the ground to the top
of a pole. The angle of elevation is 57°.
9. Find the height of the pole to the nearest tenth.
9. __________________
10. Find the distance from the base of the pole to the
10. __________________
stake to the nearest tenth.
© Glencoe/McGraw-Hill
219
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
Chapter 5, Quiz C (Lessons 5-5 and 5-6)
5
1. If 0
x
360, solve: csc x 2.
1. __________________
3 .
2. Assuming an angle in Quadrant I, evaluate tan sec1 15
2. __________________
3. Given right triangle ABC, find B to the
nearest tenth of a degree if b 7 and c 12.
3. __________________
Find each value. Round to the nearest tenth.
4. In ABC, A 58 21, C 97 07, and b 23.8. Find a.
4. __________________
5. If B 29.5, C 64.5, and a 18.8, find the area of ABC. 5. __________________
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
5
Chapter 5, Quiz D (Lessons 5-7 and 5-8)
1. Determine the number of possible solutions for ABC
if A 28, a 4, and b 11.
Find each value. Round to the nearest tenth.
2. For ABC, determine the least possible value for B
if A 49, a 12, and b 15.
1. __________________
2. __________________
3. In ABC, B 32, a 11, and c 2.4. Find b.
3. __________________
4. In ABC, a 3.1, b 5.4, and c 4.7. Find C.
4. __________________
5. If a 28.2, b 36.5, and c 40.1, find the area of ABC.
5. __________________
© Glencoe/McGraw-Hill
220
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
5
Chapter 5 SAT and ACT Practice
6. Which of the following products has the
greatest value less than 100?
After working each problem, record the
correct answer on the answer sheet
provided or use your own paper.
Multiple Choice
1. The length of a diagonal of a square
is 6 units. Find the length of a side.
units
A 92
units
B 2
C 9 units
units
D 32
E None of these
A
B
C
D
E
2. If the area of ABC is 30 square meters,
what is the length of segment CD?
A 2m
B 3m
C 5m
D 8m
E 10 m
3. The midpoint of a diameter of a circle is
(3, 4). If the coordinates of one endpoint
of the diameter are (3, 6), what are
the coordinates of the other endpoint?
A (9, 2)
B (9, 1)
C (8, 2)
D (3, 2)
E (9, 14)
4. The endpoints of a diameter of a circle
are (3, 2) and (11, 8). Find the area of
the circle.
A 5 units2
B 25 units2
C 25 units2
D 10 units2
E 5 units2
8. The diagonals of a rhombus are
perpendicular and bisect each other.
If the length of one side of a rhombus
is 25 meters and the length of one
diagonal is 14 meters, find the length
of the other diagonal.
A 7m
B 12 m
C 24 m
D 48 m
E 144 m
c 121 and b 3?
A 36
1
B 36
D 17
2
E 274
© Glencoe/McGraw-Hill
5
7. A diagonal of a rectangle is 1
inches. The length of the rectangle is
2
inches. Find the area of the
1
rectangle.
A 32
in2
B 6 in2
C 9 in2
D 65
in2
E None of these
, what is the value of a when
9. If c a1
b
5. What is the arithmetic mean of 13 and 14?
A 16
B 17
C 112
246
249
449
339
446
221
C 4
D 4
E None of these
10. How many times do the graphs of
y x2 and xy 27 intersect?
A 0
B 1
C 2
D 3
E 4
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
5
Chapter 5 SAT and ACT Practice
(continued)
13x
is an integer, then the value of x
11. If 4
CANNOT be which of the following?
A 112
B 32
C 0
D 6
E 8
16. If ƒ(x) x4 4x3 16x 16 has a
zero of 2, with a multiplicity of 3,
what is another zero of ƒ?
A 3
B 2
C 1
D 1
E 8
12. Which of the following statements
makes the expression a b represent
a negative number?
A ba
B ab
C b0
D a0
E ba
17–18. Quantitative Comparison
A if the quantity of Column A is
greater
B if the quantity in Column B is
greater
C if the two quantities are equal
D if the relationship cannot be
determined from the information
given
13. A hiker travels 5 miles due north, then
3 miles due west, and then 2 miles due
north. How far is the hiker from his
beginning point?
A About 8.2 mi
B About 7.1 mi
C About 9.4 mi
D About 6 mi
E About 7.6 mi
14. A rectangular swimming pool is
100 meters long and 25 meters wide.
Lucia swims from one corner of the
pool to the opposite corner and back
10 times. How far did she swim?
A About 193.6 m
B About 1030.8 m
C About 2061.6 m
D About 10,308.8 m
E None of these
18. The length of the hypotenuse of a
right triangle with the given leg
lengths
7 and 12
8 and 11
19. Grid-In A tent with a rectangular
floor has a diagonal length of 7 feet
and a width of 5 feet. What is the area
of the floor to the nearest square foot?
20. Grid-In Julio wants to bury a water
pipe from one corner of his field to the
opposite corner. How many feet of
pipe, to the nearest foot, does he need
if the rectangular field is 200 feet by
300 feet?
is a root of which equation?
15. 4 32
2
A x 8x 2 0
B x2 8x 18 0
C x2 16x 2 0
D x2 16x 18 0
E None of these
© Glencoe/McGraw-Hill
Column B
Column A
17. The value of x in each figure
222
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
5
Chapter 5, Cumulative Review (Chapters 1-5)
1. If ƒ(x) x 2 and g(x) 21x , find g( ƒ(x)).
1. __________________
2. Determine whether the graphs of 2x y 5 0
and x 2y 6 are parallel, coinciding, perpendicular, or
none of these.
2. __________________
3. Graph the inequality 2x 4y
3.
8.
4. Solve the following system of equations algebraically.
5x y 16
2x 3y 3
5. Find BC if B 61
6. Find the inverse of
4. __________________
3
2
5
and C .
8 4
5
5. __________________
24
6. __________________
1
, if it exists.
3
7. Determine whether the graph of y 2x4 x2 3 is
symmetric to the x-axis, the y-axis, both, or neither.
7. __________________
8. Describe the transformations relating the graph of
y x 2 5 to its parent function, y x .
8. __________________
9. Find the inverse of y 2x 6. State whether the inverse
is a function.
9. __________________
10. Solve x2 6x 6 0.
10. __________________
3
2 7 4.
11. Solve x
11. __________________
12. Find csc (90°).
12. __________________
13. Given the right triangle ABC, 13.
find side c to the nearest tenth.
14. In ABC, a 8, b 6, and c 10. Find B to the
nearest tenth.
© Glencoe/McGraw-Hill
223
14. __________________
Advanced Mathematical Concepts
Blank
NAME _____________________________ DATE _______________ PERIOD ________
SAT and ACT Practice Answer Sheet
(10 Questions)
.
/
.
/
.
.
1
2
3
4
5
6
7
8
9
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3
4
5
6
7
8
9
© Glencoe/McGraw-Hill
A1
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
SAT and ACT Practice Answer Sheet
(20 Questions)
.
/
.
/
.
.
1
2
3
4
5
6
7
8
9
0
1
2
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9
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6
7
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0
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3
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9
© Glencoe/McGraw-Hill
A2
Advanced Mathematical Concepts
© Glencoe/McGraw-Hill
57 19′ 37.2″
73.243
936
A3
30 360k;
sample answers:
690; 750
85; I
80
© Glencoe/McGraw-Hill
182
Advanced Mathematical Concepts
Santa Barbara Island: 33.476 N, 119.035 W
Santa Catalina Island: 33 23′ 9.6″ N, 118 25′ 48″ W
13. Navigation For an upcoming trip, Jackie plans to sail from
Santa Barbara Island, located at 33 28′ 32″ N, 119 2′ 7″ W, to
Santa Catalina Island, located at 33.386 N, 118.430 W. Write
the latitude and longitude for Santa Barbara Island as decimals
to the nearest thousandth and the latitude and longitude for
Santa Catalina Island as degrees, minutes, and seconds.
47
Find the measure of the reference angle for each angle.
11. 227
12. 640
112; II
If each angle is in standard position, determine a coterminal angle
that is between 0 and 360, and state the quadrant in which the
terminal side lies.
9. 472
10. 995
43 360k;
sample answers:
763; 317
Identify all angles that are coterminal with each angle. Then find
one positive angle and one negative angle that are coterminal
with each angle.
7. 43
8. 30
540
Give the angle measure represented by each rotation.
5. 1.5 rotations clockwise
6. 2.6 rotations counterclockwise
32.469
Write each measure as a decimal degree to the nearest
thousandth.
3. 32 28′ 10″
4. 73 14′ 35″
28 57′ 18″
b. p → q: If a function ƒ(x) is increasing on an
interval I, then for every a and b contained in
I, ƒ(a) ƒ(b) whenever a b. (true)
q → p: If for every a and b contained in an interval I,
ƒ(a) ƒ(b) whenever a b then function ƒ(x) is
increasing on I. (true)
Find the converse of each conditional.
a. p → q: All squares are rectangles. (true)
q → p: All rectangles are squares. (false)
© Glencoe/McGraw-Hill
183
Advanced Mathematical Concepts
functions; true. Two functions, ƒ(x) and ƒ 1(x), are inverse
functions if and only if [ ƒ ° ƒ 1](x) [ ƒ 1 ° ƒ ](x) x.
If [ ƒ ° ƒ 1](x) [ ƒ 1 ° ƒ ](x) x, then ƒ(x) and ƒ 1(x) are inverse
3. If ƒ(x) and ƒ1(x) are inverse functions, then [ ƒ ° ƒ1](x) [ ƒ1 ° ƒ](x) x.
If a function has a graph that is symmetric with respect to
the origin, then ƒ(x) ƒ(x) for all x in the domain of ƒ(x);
true. A function has a graph that is symmetric with respect
to the origin if and only if ƒ(x) ƒ(x) for all x in the
domain of ƒ(x).
2. If for all x in the domain of a function ƒ(x), ƒ(x) ƒ(x), then
the graph of ƒ(x) is symmetric with respect to the origin.
All rational numbers are integers; false
1. All integers are rational numbers.
State the converse of each conditional. Then tell if the converse is
true or false. If it is true, combine the statement and its converse
into a single statement using the words “if and only if.”
The statement “p if and only if q” means that p implies q and
q implies p.
A function ƒ(x) is increasing on an interval I if and only if for
every a and b contained in I, ƒ(a) ƒ(b) whenever a b.
In Lesson 3-5, you saw that the two statements in Example 2 can be
combined in a single statement using the words “if and only if.”
Example
If p → q is true, q → p may be either true or false.
If p and q are interchanged in a conditional statement so that p
becomes the conclusion and q becomes the hypothesis, the new
statement, q → p, is called the converse of p → q.
Enrichment
NAME _____________________________ DATE _______________ PERIOD ________
Change each measure to degrees, minutes, and seconds.
1. 28.955
2. 57.327
5-1
Reading Mathematics: If and Only If Statements
Practice
NAME _____________________________ DATE _______________ PERIOD ________
Angles and Degree Measure
5-1
Answers
(Lesson 5-1)
Advanced Mathematical Concepts
Practice
© Glencoe/McGraw-Hill
tan B 2
8
3
A4
7兹7
苶8
苶
19
; csc S ;
tan S 156
7
19兹苶
78
苶
2兹7
苶8
苶
; cot S sec S 156
7
兹1
苶0
苶
;
tan S 3; csc S 3
sec S 兹1
苶0
苶; cot S 13
© Glencoe/McGraw-Hill
about 1.419
185
Advanced Mathematical Concepts
7. Physics Suppose you are traveling in a car when a beam of light
passes from the air to the windshield. The measure of the angle
of incidence is 55, and the measure of the angle of refraction is
sin ␪i
n, to find the index of refraction n
35 15′. Use Snell’s Law, sin ␪r
of the windshield to the nearest thousandth.
2兹7
苶8
苶
;
sin S 17
; cos S 9
19
3兹1
苶0
苶
兹1
苶0
苶
; cos S ;
sin S 10
10
Find the values of the six trigonometric ratios for each ⬔S.
5.
6.
1
5
4. If sin ␪ 38, find csc ␪.
3兹5
苶5
苶
tan B 55
3. If tan ␪ 5, find cot ␪.
2兹5
苶
兹5
苶
; cos B ;
sin B 5
5
兹5
苶5
苶
;
sin B 38; cos B 8
Find the values of the sine, cosine, and tangent for each ⬔B.
1.
2.
Trigonometric Ratios in Right Triangles
5-2
NAME _____________________________ DATE _______________ PERIOD ________
1
1
© Glencoe/McGraw-Hill
186
35
18.5
Advanced Mathematical Concepts
Find the area of the triangle having vertices with each set of coordinates.
2. A(–2, –4), B(4, 7), C(6, –1)
1. A(4, 6), B(–1, 2), C(6, –5) 31.5
3. A(4, 2), B(6, 9), C(–1, 4) 19.5
4. A(2, –3), B(6, –8), C(3, 5)
17.5 square units
Area 䉭ABC 5(9) (5)(5) (4)(3) (2)(9)
2
2
2
1
Area 䉭ABC area ADEF area 䉭ADB area 䉭BEC area 䉭CFA, where 䉭ADB, 䉭BEC,
and 䉭CFA are all right triangles.
A rectangle is placed around the triangle so that the vertices of the
triangle all touch the sides of the rectangle.
Example
Find the area of a triangle whose vertices are
A(1, 3), B(4, 8), and C(8, 5).
Plot the points and draw the triangle. Encase the triangle
in a rectangle whose sides are parallel to the
axes, then find the coordinates of the vertices of the
rectangle.
The vertices of a triangle can be represented on the coordinate plane
by three ordered pairs. In order to find the area of a general triangle,
you can encase the triangle in a rectangle as shown in the diagram
below.
the base, and the other leg can be used as the height.
A bh. In the formula, one leg of the right triangle can be used as
2
1
You can find the area of a right triangle by using the formula
Enrichment
Using Right Triangles to Find the Area of
Another Triangle
5-2
NAME _____________________________ DATE _______________ PERIOD ________
Answers
(Lesson 5-2)
Advanced Mathematical Concepts
Practice
© Glencoe/McGraw-Hill
undefined
A5
cot undefined
cot 15
188
sec 1
sec 兹2
苶6
苶
© Glencoe/McGraw-Hill
csc undefined
兹2
苶
苶
6
csc 5
Advanced Mathematical Concepts
cot 43
sec 53
csc 54
tan 34
cos 35
cos 1
tan 0
sin 45
sin 0
tan 5
兹2
苶6
苶
cos 26
sin 26
5兹2
苶6
苶
Find the values of the six trigonometric functions for angle in
standard position if a point with the given coordinates lies on its
terminal side.
6. (1, 5)
7. (7, 0)
8. (3, 4)
兹
3
苶
苶 cot 120 tan 120 兹3
3
24
28
32
1000
6.
7.
8.
2
3.1415720
3.1214452
3.1152931
3.1058285
3.0901699
3
2.8284271
0.0000207
0.0201475
0.0262996
0.0357641
0.0514227
0.1415927
0.3131655
1.1415927
1.8425545
Area of Circle
less
Area of Polygon
3.1416030
3.1517249
3.1548423
3.1596599
3.1676888
3.2153903
3.3137085
4
5.1961524
Area of
Circumscribed
Polygon
0.0000103
0.0101323
0.0132497
0.0180673
0.0260962
0.0737977
0.1721158
0.8584073
2.0545598
Area of Polygon
less
Area of Circle
© Glencoe/McGraw-Hill
189
Advanced Mathematical Concepts
9. What number do the areas of the circumscribed and inscribed
polygons seem to be approaching as the number of sides of the
polygon increases?
20
5.
12
3.
4.
8
2.
4
1.2990381
3
1.
sec 120 2
Area of
Inscribed
Polygon
Number
of Sides
Use a calculator to complete the chart below for a unit circle (a circle of radius 1).
cos 120 12
2兹3
苶
180°
nr2
360°
AI 2 sin n
AC ␲ r2
Area of circumscribed polygon AC nr2 tan n
Area of inscribed polygon
Area of circle
A regular polygon has sides of equal length and angles of equal
measure. A regular polygon can be inscribed in or circumscribed
about a circle. For n-sided regular polygons, the following area
formulas can be used.
csc 120 3
cot 45 1
sec 45 兹2
苶
csc 45 兹2
苶
3. sin (90)
NAME _____________________________ DATE _______________ PERIOD ________
Enrichment
Areas of Polygons and Circles
5-3
兹苶
3
sin 120 2
5. 120
tan 45 1
兹2
苶
sin 45 2
兹2
苶
cos 45 2
Use the unit circle to find the values of the six trigonometric
functions for each angle.
4. 45
1
Use the unit circle to find each value.
1. csc 90
2. tan 270
1
NAME _____________________________ DATE _______________ PERIOD ________
Trigonometric Functions on the Unit Circle
5-3
Answers
(Lesson 5-3)
Advanced Mathematical Concepts
Practice
© Glencoe/McGraw-Hill
A6
© Glencoe/McGraw-Hill
about 102.3 ft
191
8. Observation A person standing 100 feet from the bottom
of a cliff notices a tower on top of the cliff. The angle of
elevation to the top of the cliff is 30. The angle of elevation
to the top of the tower is 58. How tall is the tower?
c. What is the perimeter of the hexagon? about 33 cm
Advanced Mathematical Concepts
b. What is the length of a side of the hexagon? about 5.5 cm
a. Find the radius of the circumscribed circle. about 5.5 cm
7. Geometry A circle is circumscribed about a regular hexagon with an
apothem of 4.8 centimeters.
about 10.2 ft
b. What is the horizontal distance between the
bottom of the ladder and the house?
about 28.2 ft
6. Construction A 30-foot ladder leaning against
the side of a house makes a 70 5′ angle with the
ground.
a. How far up the side of the house does the
ladder reach?
14.6
5. If b 14 and A 16, find c.
9.4
4. If B 64 and b 19.2, find a.
52.8
3. If B 56 48′ and c 63.1, find b.
10.4
2. If a 9 and B 49, find b.
13.3
Solve each problem. Round to the nearest tenth.
1. If A 55 55′ and c 16, find a.
Applying Trigonometric Functions
5-4
NAME _____________________________ DATE _______________ PERIOD ________
© Glencoe/McGraw-Hill
192
2. your school’s flagpole
3. a tree on your school’s property
4. the highest point on the front wall of your school building
5. the goal posts on a football field
6. the hoop on a basketball court
7. the top of the highest window of your school building
8. the top of a school bus
9. the top of a set of bleachers at your school
10. the top of a utility pole near your school
See students’ work.
Advanced Mathematical Concepts
Use your hypsometer to find the height of each of the following.
See students’ diagrams.
1. Draw a diagram to illustrate how you can use similar triangles
and the hypsometer to find the height of a tall object.
Sight the top of the object through the straw. Note where the freehanging string crosses the bottom scale. Then use similar triangles
to find the height of the object.
To use the hypsometer, you will need to measure the distance from
the base of the object whose height you are finding to where you
stand when you use the hypsometer.
Mark off 1-cm increments along one short side and one long side of
the cardboard. Tape the straw to the other short side. Then attach
the weight to one end of the string, and attach the other end of the
string to one corner of the cardboard, as shown in the figure below.
The diagram below shows how your hypsometer should look.
A hypsometer is a device that can be used to measure the height of
an object. To construct your own hypsometer, you will need a
rectangular piece of heavy cardboard that is at least 7 cm by 10 cm,
a straw, transparent tape, a string about 20 cm long, and a small
weight that can be attached to the string.
Enrichment
Making and Using a Hypsometer
5-4
NAME _____________________________ DATE _______________ PERIOD ________
Answers
(Lesson 5-4)
Advanced Mathematical Concepts
Practice
© Glencoe/McGraw-Hill
45, 225
45, 315
冢
冣
兹5
苶
2
5. tan 冢cos1 23冣
A7
© Glencoe/McGraw-Hill
194
Advanced Mathematical Concepts
12
13
6. cos 冢arcsin 15
3冣
13. Recreation The swimming pool at Perris Hill Plunge is 50 feet
long and 25 feet wide. The bottom of the pool is slanted so that
the water depth is 3 feet at the shallow end and 15 feet at the
deep end. What is the angle of elevation at the bottom of the
pool? about 13.5
c 7.3, A 15.9, B 74.1
12. a 2, b 7
a 3.9, b 13.5, B 74
11. A 16, c 14
A 41, b 10.4, c 13.7
Solve each triangle described, given the triangle at the right.
Round to the nearest tenth, if necessary.
10. a 9, B 49
41.4
9. If q 20 and r 15, find S.
71.6
8. If r 12 and s 4, find R.
17.5
Solve each problem. Round to the nearest tenth.
7. If q 10 and s 3, find S.
3
兹苶
3
兹
苶
3
4. tan tan1 3
1
30, 150
Evaluate each expression. Assume that all angles are in Quadrant I.
2. tan x 1
1. cos x 2
兹2
苶
Solve each equation if 0 x 360.
3. sin x 2
NAME _____________________________ DATE _______________ PERIOD ________
Solving Right Triangles
5-5
Enrichment
1
5
21.8°
m⬔MAB Arctan
© Glencoe/McGraw-Hill
195
Advanced Mathematical Concepts
The tangent function is not linear. The ratio of the measures
of two angles is not equal to the ratio of the tangents of the
angles.
6. Find m⬔CAB and use it to find m ⬔1. m⬔1 ⬇ 9.2°
7. Explain why the proposed method for trisecting an angle fails.
4. Find the measure of ⬔MAB. m⬔MAB ⬇
5. Find the measure of ⬔2. m⬔2 ⬇ 10.5°
3. Write m⬔MAB as the value of an inverse function.
m⬔3 ⬇ 11.3°
2. Find the measure of ⬔3.
m⬔3 Arctan
1. Express m⬔3 as the value of an inverse function.
The proposed method has been used to construct the figure below.
CM MN NB 1. AB 5. Follow the instructions to show that
the three angles ⬔1, ⬔2, and ⬔3, are not congruent. Find angle
measures to the nearest tenth of a degree.
Given: ⬔ A
Claim: ⬔ A can be trisected using the following method.
Method: Choose point C on one ray of ⬔ A .
Through C construct a perpendicular to
the other ray, intersecting it at B.
Construct M and N, the points that
苶 into three congruent
divide 苶
CB
segments. Draw 苶
AM
苶 and 苶
AN
苶, which
trisect ⬔CAB into the congruent angles
⬔1, ⬔2, and ⬔3.
Most geometry texts state that it is impossible to trisect an arbitrary
angle using only a compass and straightedge. This fact has been
known since ancient times, but since it is usually stated without
proof, some geometry students do not believe it. If the students set
out to find a method for trisecting angles, they will probably try the
following method. It is based on the familiar construction which
allows a segment to be divided into any desired number of congruent
segments. You can use inverse trigonometric functions to show that
application of the method to the trisection of angles is not valid.
2
5
NAME _____________________________ DATE _______________ PERIOD ________
Disproving Angle Trisection
5-5
Answers
(Lesson 5-5)
Advanced Mathematical Concepts
© Glencoe/McGraw-Hill
C 118, a 46.1, c 74.7
C 105, b 14.1, c 19.3
4. A 30, B 45, a 10
A8
2526.7 units2
© Glencoe/McGraw-Hill
about 25.9 ft
197
Advanced Mathematical Concepts
10. a 68, c 110, B 42.5
154.1 units2
8. b 14, C 110, B 25
0.7 units2
11. Street Lighting A lamppost tilts toward the sun
at a 2 angle from the vertical and casts a 25-foot
shadow. The angle from the tip of the shadow to the
top of the lamppost is 45. Find the length of the
lamppost.
135.9 units2
9. b 15, c 20, A 115
64.3 units2
7. A 50, b 12, c 14
4.7 units2
Find the area of each triangle. Round to the nearest tenth.
5. c 4, A 37, B 69
6. C 85, a 2, B 19
B 10, b 69.5,
c 136.8
3. A 150, C 20, a 200
C 79, a 9.4, b 13.6
Solve each triangle. Round to the nearest tenth.
1. A 38, B 63, c 15
2. A 33, B 29, b 41
Practice
The Law of Sines
5-6
NAME _____________________________ DATE _______________ PERIOD ________
Enrichment
G
a
e
J
b
c
198
b ⬇ 113.4; c ⬇ 110.8; e ⬇ 52.3; f ⬇ 40.5;
A ⬇ 32°37'; ; C ⬇ 71°23'; D ⬇ 55°37';
E ⬇ 82°23'; J ⬇ 100°
f
a ⬇ 66.7; b ⬇ 52.7; e ⬇ 49.5
A ⬇ 103°58'; B ⬇ 50°8'; D ⬇ 76°8';
E ⬇ 50°52'; G ⬇ 76°52'; H ⬇ 54°8'
H
e
© Glencoe/McGraw-Hill
2.
1.
Advanced Mathematical Concepts
A surveyor took the following measurements from two irregularlyshaped pieces of land. Some of the lengths and angle measurements
are missing. Find all missing lengths and angle measurements.
Round lengths to the nearest tenth and angle measurements to the
nearest minute.
Triangle Challenge
5-6
NAME _____________________________ DATE _______________ PERIOD ________
Answers
(Lesson 5-6)
Advanced Mathematical Concepts
Practice
NAME _____________________________ DATE _______________ PERIOD ________
© Glencoe/McGraw-Hill
0
4. A 110, a 4, c 4
0
A9
A 67.9, B 62.1, a 15.7;
A 12.1, B 117.9, a 3.6
10. b 15, c 13, C 50
none
8. a 12, b 15, A 55
B 30.5, C 124.5, c 243.7;
B 149.5, C 5.5, c 28.3
6. a 125, A 25, b 150
© Glencoe/McGraw-Hill
about 312.1 ft
200
Advanced Mathematical Concepts
11. Property Maintenance The McDougalls plan to fence a triangular parcel of their
land. One side of the property is 75 feet in length. It forms a 38 angle with another
side of the property, which has not yet been measured. The remaining side of the
property is 95 feet in length. Approximate to the nearest tenth the length of fence
needed to enclose this parcel of the McDougalls’ lot.
B 21.4, C 116.6,
c 29.4
9. A 42, a 22, b 12
B 32.6, C 35.4,
b 18.6
7. a 32, c 20, A 112
none
5. b 50, a 33, A 132
Find all solutions for each triangle. If no solutions exist, write
none. Round to the nearest tenth.
2
3. A 58, a 4.5, b 5
1
Determine the number of possible solutions for each triangle.
1. A 42, a 22, b 12
2. a 15, b 25, A 85
The Ambiguous Case for the Law of Sines
5-7
NAME _____________________________ DATE _______________ PERIOD ________
sin 72°
sin 59°
sin 105°
201
a 60°, B 121°, C 45°
© Glencoe/McGraw-Hill
Advanced Mathematical Concepts
4. b 94°, c 55°, A 48°
A 59°, B 124°, c 59°
3. a 76°, b 110°, C 49°
A 116°, B 31°, C 71°
2. a 110°, b 33°, c 97°
A 41°, B 39°, C 128°
1. a 56°, b 53°, c 94°
sin 61°
1.1
sin 119°
sin 52°
Check by using the Law of Sines.
0.4848 (0.3090)(–0.2588) (0.9511)(0.9659) cos C
cos C 0.6148
C 52°
–0.2588 (0.3090)(0.4848) (0.9511)(0.8746) cos B
cos B –0.4912
B 119°
0.3090 (–0.2588)(0.4848) (0.9659)(0.8746) cos A
cos A 0.5143
A 59°
Use the Law of Cosines.
Solve the spherical triangle given a 72°, b 105°, and c 61°.
Solve each spherical triangle.
Example
There is also a Law of Cosines for spherical triangles.
cos a cos b cos c sin b sin c cos A
cos b cos a cos c sin a sin c cos B
cos c cos a cos b sin a sin b cos C
sin a
sin b
sin c
sin A
sin B
sin C
The sum of the sides of a spherical triangle is less than 360°.
The sum of the angles is greater than 180° and less than 540°.
The Law of Sines for spherical triangles is as follows.
Spherical trigonometry is an extension of plane trigonometry.
Figures are drawn on the surface of a sphere. Arcs of great
circles correspond to line segments in the plane. The arcs of
three great circles intersecting on a sphere form a spherical
triangle. Angles have the same measure as the tangent lines
drawn to each great circle at the vertex. Since the sides are
arcs, they too can be measured in degrees.
Enrichment
Spherical Triangles
5-7
Answers
(Lesson 5-7)
Advanced Mathematical Concepts
© Glencoe/McGraw-Hill
b 13.8, A 45.5, C 34.5
A10
10.4 units2
8. a 8, b 7, c 3
71.0 units2
© Glencoe/McGraw-Hill
4.1 mi
203
10. Orienteering During an orienteering hike, two hikers
start at point A and head in a direction 30 west of south
to point B. They hike 6 miles from point A to point B. From
point B, they hike to point C and then from point C back to
point A, which is 8 miles directly north of point C. How
many miles did they hike from point B to point C?
about 28.3
9. The sides of a triangle measure 13.4 centimeters,
18.7 centimeters, and 26.5 centimeters. Find the measure
of the angle with the least measure.
33.7 units2
7. a 14, b 9, c 8
30.0 units2
Advanced Mathematical Concepts
A 27.7, B 40.5, C 111.8
4. a 5, b 7, c 10
Find the area of each triangle. Round to the nearest tenth.
5. a 5, b 12, c 13
6. a 11, b 13, c 16
a 40.5, B 52.0,
C 75.0
3. c 49, b 40, A 53
A 38.2, B 21.8,
C 120.0
Solve each triangle. Round to the nearest tenth.
1. a 20, b 12, c 28
2. a 10, c 8, B 100
Practice
The Law of Cosines
5-8
NAME _____________________________ DATE _______________ PERIOD ________
0, y 2 a 2 b 2
© Glencoe/McGraw-Hill
204
min (19 7); max (19 7); no
Advanced Mathematical Concepts
c. How do the answers for part b relate to the lengths 7 and 19?
Are the maximum and minimum values from part b ever
actually attained in the geometric situation?
See students’ graphs;
Ymin 12, Ymax 26
b. Display the graph and use the TRACE function. What do the
maximum and minimum values appear to be for the function?
Xmin 0; Xmax 180
a. In view of the geometry of the situation, what range of values
should you use for X on a graphing calculator?
See students’ graphs.
5. Consider some particular values of a and b, say 7 for a and
19 for b. Use a graphing calculator to graph the equation you
get by solving y2 72 192 2(7)(19) cos (x°) for y.
decreases to –1
4. What happens to the value of cos (x°) as x increases beyond 90
and approaches 180?
3. If x equals 90, what is the value of cos (x°)? What does the
equation of y2 a2 b2 2ab cos (x°) simplify to if x equals 90?
decreases, approaches 0
2. If the value of x is very close to zero but then increases, what
happens to cos (x°) as x approaches 90?
1
1. If the value of x becomes less and less, what number is cos (x°)
close to?
Answer the following questions to clarify the relationship between
the Law of Cosines and the Pythagorean Theorem.
y2 a2 b2 2ab cos (x°).
The law of cosines bears strong similarities to the
Pythagorean Theorem. According to the Law of
Cosines, if two sides of a triangle have lengths a and
b and if the angle between them has a measure of x,
then the length, y, of the third side of the triangle
can be found by using the equation
Enrichment
The Law of Cosines and the Pythagorean Theorem
5-8
NAME _____________________________ DATE _______________ PERIOD ________
Answers
(Lesson 5-8)
Advanced Mathematical Concepts
Chapter 5 Answer Key
Form 1A
Page 205
1.
B
2.
D
3.
A
4.
A
5.
C
6.
B
7.
8.
9.
D
A
Form 1B
Page 206
14.
B
15.
A
16.
C
17.
B
Page 207
1.
D
2.
A
Page 208
14.
A
15.
B
3.
D
4.
C
16.
A
5.
D
17.
A
6.
B
18.
A
19.
C
18.
D
19.
B
20.
D
20.
C
21.
A
21.
B
22.
B
23.
B
22.
7.
D
8.
B
9.
B
10.
A
C
C
C
10.
D
23.
11.
C
24.
D
11.
C
24.
C
12.
A
25.
D
12.
D
25.
D
Bonus:
13.
B
Bonus:
B
© Glencoe/McGraw-Hill
13.
A11
A
C
Advanced Mathematical Concepts
Chapter 5 Answer Key
Form 1C
Page 209
1.
A
2.
B
3.
A
4.
C
5.
B
6.
B
Form 2A
Page 210
A
8.
D
9.
D
10.
11.
12.
D
B
A
C
1. 225° 38 20.4
14. 179 feet
15.
A
2. 23.274°
15. 48 feet
16.
B
3.
17.
19.
B
21.
C
22.
B
23.
D
25.
D
C
16. 150° and 330°
22°
6.
4兹
6
苶
11
17.
5
4
18.
68.2°
19.
17.6
7.
5兹
6
苶
24
8.
11
5
21.
two
9.
12
22.
3.9
10.
1
兹3
苶
23.
12.0
11.
24.
22.2°
12.
兹1
苶
苶
3
3
13.
153
20. 103.7 units2
25. 136.8 units2
B
Bonus:
A
© Glencoe/McGraw-Hill
5.
B
C
24.
864°
4.180° and 180°
A
20.
Bonus:
13.
Page 212
14.
18.
7.
Page 211
A12
1.054
Advanced Mathematical Concepts
Chapter 5 Answer Key
Form 2B
Form 2C
Page 214
Page 213
1. 124° 37 48
14. 186 feet
2. 48.538°
15. 135 feet
3.
450°
5.
7.
8.
2. 75.500°
3.
18.
540°
12
13
4.
兹3
苶
苶
3
7
7
4
10.7
20. 164.9 units2
21.
5.
55°
6.
兹6
苶
苶
5
9
7.
4兹
6
苶5
苶
65
8.
9
4
9.
13
one
兹3苶3
苶
4
15. 75 feet
16. 240° and 300°
17.
5
13
18.
41.8°
19.
19.8
22.
10.
0
23.
10.5
2
24.
129.8°
11.
10.
0
11.
兹
3
苶
2
12.
4兹
1
苶7
苶
17
21.
none
22.
19.8
23.
9.3
24.
117.3
25. 196.7 units2
25. 154.7 units2
兹4苶1
苶
4
20. 120.7 units2
9.9
14
13.
14. 185 feet
135°
56.9°
9.
12.
Page 216
50°
19.
6.
1. 25° 36 00
16. 210° and 330°
17.
4. 90° and 270°
Page 215
Bonus: 0.9487
13.
3
15
Bonus:
1.000
152
© Glencoe/McGraw-Hill
A13
Advanced Mathematical Concepts
Chapter 5 Answer Key
CHAPTER 5 SCORING RUBRIC
Level
Specific Criteria
3 Superior
• Shows thorough understanding of the concepts standard
position, degree measure, quadrant, reference angle,
and the six trigonometric functions of an angle.
• Computations are correct.
• Uses appropriate strategies to solve problems.
• Written explanations are exemplary.
• Goes beyond requirements of some or all problems.
2 Satisfactory,
with Minor
Flaws
• Shows understanding of the concepts standard position,
degree measure, quadrant, reference angle, and the six
trigonometric functions of an angle.
• Uses appropriate strategies to solve problems.
• Computations are mostly correct.
• Written explanations are effective.
• Satisfies all requirements of problems.
1 Nearly
Satisfactory,
with Serious
Flaws
• Shows understanding of most of the concepts standard
position, degree measure, quadrant, reference angle,
and the six trigonometric functions of an angle.
• May not use appropriate strategies to solve problems.
• Computations are mostly correct.
• Written explanations are satisfactory.
• Satisfies most requirements of problems.
0 Unsatisfactory • Shows little or no understanding of the concepts
standard position, degree measure, quadrant, reference
angle, and the six trigonometric functions of an angle.
• May not use appropriate strategies to solve problems.
• Computations are incorrect.
• Written explanations are not satisfactory.
• Does not satisfy requirements of problems.
© Glencoe/McGraw-Hill
A14
Advanced Mathematical Concepts
Chapter 5 Answer Key
Open-Ended Assessment
Page 217
2a. The length of the missing side
1a. 210°, 570°, 930°
to the east is given by
3
1b. The cosine is x , or
苶
r
2兹3
15
ft.
sin 40°
兹
苶.
3
2
The length of the missing side
around the fountain is given
by 16 20, or 10.5 feet.
1c. first quadrant: 30°,
second quadrant: 150°,
fourth quadrant: 330°
The total length is
approximately 140.7 feet.
1d. Sample answers:
(3, 4); sin A 45 ,
cos A 35, tan A 43,
2b. Answers will vary but might
include issues such as the
length of fence required versus
the increase in
park size, access and/or
proximity to roads,
maintenance and/or
enhancement of the
fountain, and so on.
csc A 54, sec A 53,
cot A 34
© Glencoe/McGraw-Hill
x , or 20.2 feet.
sin 60°
A15
Advanced Mathematical Concepts
Chapter 5 Answer Key
Mid-Chapter Test
Page 218
1.
2.
65° 46 55.2
250°; III
Quiz A
Page 219
1.
2.
3.
47° 16 59
122.720°
810°
4. 120° and 240°
3.
4.
5.
6.
7.
8.
9.
10.
2兹
1
苶0
苶
11
5.
2兹
1
苶苶
0
9
7.
13
6.
1.
210° and 330°
2.
12
5
3.
35.7°
4.
48.8
5.
78.7 units2
37°
3兹
3
苶4
苶
34
3
5
8.
兹3
苶
苶
4
5
9.
32
10.
Quiz C
Page 220
2
0
兹
3
苶
2
兹5
苶
Quiz B
Page 219
1.
1
2.
3.
undefined
4.
5.
兹
3
苶
2
1
Quiz D
Page 220
兹苶
4
苶
1
4
6.
5
13
7.
8.
6.6
9.8
1.
none
2.
70.6°
3.
9.1
4.
60.1°
5.
498.0 units2
9.2
11.0
9.
10.
© Glencoe/McGraw-Hill
83.9 feet
54.5 feet
A16
Advanced Mathematical Concepts
Chapter 5 Answer Key
SAT/ACT Practice
Page 221
1.
2.
D
D
Cumulative Review
Page 222
11.
12.
D
Page 223
1.
1
2x 4
2.
perpendicular
A
3.
3.
4.
5.
A
C
E
13.
14.
15.
E
C
4.
(3, 1)
5.
38
冤 34
37 15冥
6.
A
7.
6.
E
16.
8.
B
D
17.
18.
B
B
A
9.
D
19.
24
10.
B
20.
361
© Glencoe/McGraw-Hill
冥
y-axis
reflected over x-axis;
translated left 2 units
8. and down 5 units
9.
7.
冤
3 1
10 10
1
2
5
5
A17
x 6 ; yes
y
2
10.
3 兹3
苶
11.
29
12.
1
13.
15.4
14.
36.9°
Advanced Mathematical Concepts
BLANK