NAME ______________________________________________ DATE
8
____________ PERIOD _____
Reading to Learn Mathematics
This is an alphabetical list of the key vocabulary terms you will learn in Chapter 13.
As you study the chapter, complete each term’s definition or description. Remember
to add the page number where you found the term. Add these pages to your Algebra
Study Notebook to review vocabulary at the end of the chapter.
Vocabulary Term
Found
on Page
Definition/Description/Example
angle of depression
or elevation
⎧
⎪
⎨
⎪
⎩
Arccosine function
AHRK·KOH·SYN
⎧
⎪
⎨
⎪
⎩
Arcsine function
AHRK·SYN
⎧
⎪
⎪
⎨
⎪
⎪
⎩
Arctangent function
AHRK·TAN·juhnt
⎧
⎪
⎨
⎪
⎩
cosecant
KOH·SEE·KANT
cosine
coterminal angles
cotangent
Law of Cosines
Law of Sines
(continued on the next page)
© Glencoe/McGraw-Hill
vii
Vocabulary Builder
Vocabulary Builder
NAME ______________________________________________ DATE
8
____________ PERIOD _____
Reading to Learn Mathematics
Vocabulary Builder (continued)
Vocabulary Term
Found
on Page
Definition/Description/Example
period
principal values
⎧
⎪
⎪
⎨
⎪
⎪
⎩
quadrantal angles
kwah·DRAN·tuhl
⎧
⎪
⎨
⎪
⎩
radian
RAY·dee·uhn
reference angle
secant
sine
standard position
tangent
⎧
⎪
⎪
⎨
⎪
⎪
⎩
trigonometry
TRIH·guh·NAH·muh·tree
© Glencoe/McGraw-Hill
viii
NAME ______________________________________________ DATE
8
____________ PERIOD _____
Reading to Learn Mathematics
This is an alphabetical list of the key vocabulary terms you will learn in Chapter 14.
As you study the chapter, complete each term’s definition or description. Remember
to add the page number where you found the term. Add these pages to your Algebra
Study Notebook to review vocabulary at the end of the chapter.
Vocabulary Term
Found
on Page
Definition/Description/Example
⎧
⎪
⎪
⎨
⎪
⎪
⎩
amplitude
AM·pluh·TOOD
double-angle formula
half-angle formula
midline
⎧
⎨
⎩
phase shift
FAYZ
(continued on the next page)
© Glencoe/McGraw-Hill
vii
Vocabulary Builder
Vocabulary Builder
NAME ______________________________________________ DATE
8
____________ PERIOD _____
Reading to Learn Mathematics
Vocabulary Builder (continued)
Vocabulary Term
Found
on Page
Definition/Description/Example
trigonometric equation
trigonometric identity
vertical shift
© Glencoe/McGraw-Hill
viii
Glencoe Algebra 2
NAME ______________________________________________ DATE
____________ PERIOD _____
Study Guide and Intervention
Right Triangle Trigonometry
Trigonometric Values
If ␪ is the measure of an acute angle of a right triangle, opp is the measure of the
leg opposite ␪, adj is the measure of the leg adjacent to ␪, and hyp is the measure
of the hypotenuse, then the following are true.
Trigonometric Functions
B
hyp
A
␪
adj
opp
adj
opp
sin ␪ ⫽ ᎏ
opp
hyp
cos ␪ ⫽ ᎏ
adj
hyp
tan ␪ ⫽ ᎏ
C
csc ␪ ⫽ ᎏ
hyp
opp
sec ␪ ⫽ ᎏ
hyp
adj
cot ␪ ⫽ ᎏ
adj
opp
Example
Find the values of the six trigonometric functions for angle ␪.
Use the Pythagorean Theorem to find x, the measure of the leg opposite ␪.
7
␪
2
2
2
Pythagorean Theorem
x ⫹7 ⫽9
x2 ⫹ 49 ⫽ 81
Simplify.
9
x2 ⫽ 32
Subtract 49 from each side.
x ⫽ 兹苶
32 or 4兹苶
2
Take the square root of each side.
Use opp ⫽ 4兹2
苶, adj ⫽ 7, and hyp ⫽ 9 to write each trigonometric ratio.
4兹2
苶
7
9
sin ␪ ⫽ ᎏ
9
cos ␪ ⫽ ᎏ
4兹2
苶
9兹2
苶
tan ␪ ⫽ ᎏ
7
7兹2
苶
9
7
csc ␪ ⫽ ᎏ
8
x
sec ␪ ⫽ ᎏ
cot ␪ ⫽ ᎏ
8
Exercises
Find the values of the six trigonometric functions for angle ␪.
1.
2.
5
␪
3.
8
13
12
5
13
13
5
13
tan
17 ␪ 12 ; csc ␪ 5 ;
;
8
13
12
sec ␪ ; cot ␪ 12
5
␪
4
3
5
5
4
5
tan ␪ ; csc ␪ ;
3
4
15
8
17
17
8
tan ␪ ; csc ␪ 15
5
3
sin ␪ ; cos ␪ ;
3
4
17
15
sec ␪ ; cot ␪ 5.
9
␪
16
12
sin ␪ ; cos ␪ ; sin ␪ ; cos ␪ ;
4.
17
␪
6.
␪
6
3
9
15
8
sec ␪ ; cot ␪ 10
␪
12
兹2
苶
sin ␪ ; cos ␪ 3
兹苶
1
2
5兹61
苶
61
2
2
兹苶
; tan ␪ 1; csc ␪ 2
sin ␪ ; cos ␪ ;
sin ␪ ; cos ␪ tan ␪ 兹3
苶;
兹2
苶; sec ␪ 兹2
苶;
csc ␪ ;
61
5
6兹苶
; tan ␪ ;
6
61
61
兹苶
csc ␪ ; sec ␪ 5
© Glencoe/McGraw-Hill
2
2兹3
苶
3
775
NAME ______________________________________________ DATE
1
____________ PERIOD _____
Study Guide and Intervention (continued)
Right Triangle Trigonometry
Right Triangle Problems
Example
Solve 䉭ABC. Round measures of sides to the
nearest tenth and measures of angles to the nearest degree.
You know the measures of one side, one acute angle, and the right angle.
You need to find a, b, and A.
A
18
b
Find a and b.
b
18
a
18
sin 54⬚ ⫽ ᎏᎏ
B
cos 54⬚ ⫽ ᎏᎏ
b ⫽ 18 sin 54⬚
b ⬇ 14.6
54
a
C
a ⫽ 18 cos 54⬚
a ⬇ 10.6
Find A.
54⬚ ⫹ A ⫽ 90⬚
A ⫽ 36⬚
Angles A and B are complementary.
Solve for A.
Therefore A ⫽ 36⬚, a ⬇ 10.6, and b ⬇ 14.6.
Exercises
Write an equation involving sin, cos, or tan that can be used to find x. Then solve
the equation. Round measures of sides to the nearest tenth.
1.
2.
3.
63
10
38
x
10
x
tan 38 ; 12.8
4
14.5
x
x
20
4
x
cos 63 ; 8.8
x
14.5
sin 20 ; 5.0
Solve 䉭ABC by using the given measurements. Round measures
of sides to the nearest tenth and measures of angles to the
nearest degree.
A
C
4. A ⫽ 80⬚, b ⫽ 6
5. B ⫽ 25⬚, c ⫽ 20
6. b ⫽ 8, c ⫽ 14
7. a ⫽ 6, b ⫽ 7
8. a ⫽ 12, B ⫽ 42⬚
9. a ⫽ 15, A ⫽ 54⬚
a ⬇ 34.0, c ⬇ 34.6,
B 10
c ⬇ 9.2, A ⬇ 41,
B ⬇ 49
© Glencoe/McGraw-Hill
a ⬇ 18.1, b ⬇ 8.5,
A 65
b ⬇ 10.8, c ⬇ 16.1,
A 48
776
c
b
a
a ⬇ 11.5, B ⬇ 35,
C ⬇ 55
b ⬇ 10.9, c ⬇ 18.5,
B 36
B
NAME ______________________________________________ DATE
____________ PERIOD _____
Skills Practice
Right Triangle Trigonometry
Find the values of the six trigonometric functions for angle ␪.
1.
2.
␪
3.
5
2
6
␪
8
4
3
5
5
4
5
tan ␪ , csc ␪ ,
3
4
5
3
sec ␪ , cot ␪ 3
4
3
13
12
5
13
13
13
5
tan ␪ , csc ␪ ,
5
12
13
12
sec ␪ , cot ␪ 12
5
sin ␪ , cos ␪ ,
␪
13
3兹苶
13
13
2兹苶
cos ␪ ,
13
3
兹13
苶
tan ␪ , csc ␪ ,
2
3
13
2
兹苶
sec ␪ , cot ␪ 3
2
sin ␪ , cos ␪ ,
sin ␪ ,
Write an equation involving sin, cos, or tan that can be used to find x. Then solve
the equation. Round measures of sides to the nearest tenth and measures of
angles to the nearest degree.
4.
5.
6.
60
8
x
x
5
10
22
30
x
8
x
7.
cos 60 , x 10
8.
60
x
10
5
x
tan 30 , x ⬇ 13.9
5
9.
x
8
tan 22 , x ⬇ 4.0
x
2
5
4
x
x
5
sin 60 , x ⬇ 4.3
5
8
cos x , x ⬇ 51
4
2
tan x , x ⬇ 63
Solve 䉭ABC by using the given measurements. Round measures of
sides to the nearest tenth and measures of angles to the nearest degree.
10. A ⫽ 72⬚, c ⫽ 10
a ⬇ 9.5, b ⬇ 3.1, B 18
11. B ⫽ 20⬚, b ⫽ 15
a ⬇ 41.2, c ⬇ 43.9, A 70
12. A ⫽ 80⬚, a ⫽ 9
13. A ⫽ 58⬚, b ⫽ 12
14. b ⫽ 4, c ⫽ 9
15. a ⫽ 7, b ⫽ 5
b ⬇ 1.6, c ⬇ 9.1, B 10
a ⬇ 8.1, A ⬇ 64, B ⬇ 26
© Glencoe/McGraw-Hill
a ⬇ 19.2, c ⬇ 22.6, B 32
c ⬇ 8.6, A ⬇ 54, B ⬇ 36
777
A
b
C
c
a
B
NAME ______________________________________________ DATE
____________ PERIOD _____
Practice (Average)
Right Triangle Trigonometry
Find the values of the six trigonometric functions for angle ␪.
1.
2.
3.
3兹苵
3
␪
␪
5
45
␪
3
11
24
15
1
8
5
4兹苶6
兹3
苶
sin ␪ , cos ␪ ,
17
2
17
11
2
11
15
17
11
5兹6
苶
兹3
苶
tan ␪ , csc ␪ 2,
tan ␪ , csc ␪ , tan ␪ , csc ␪ ,
8
15
5
3
24
6
3
17
8
11兹6
4
2
苶
兹苶
兹苶
sec ␪ , cot ␪ sec ␪ , cot ␪ sec ␪ , cot ␪ 8
15
5
3
24
sin ␪ , cos ␪ , sin ␪ , cos ␪ ,
兹苶
3
Write an equation involving sin, cos, or tan that can be used to find x. Then solve
the equation. Round measures of sides to the nearest tenth and measures of
angles to the nearest degree.
4.
5.
6.
49
x
x
x
17
30
32
7
x
7
sin 20 , x ⬇ 10.9
8.
tan 49 , x ⬇ 14.8
9.
7
x
19.2
x
41
28
17
x
x
32
tan 30 , x ⬇ 4.0
7.
20
28
x
cos 41 , x ⬇ 37.1
17
x
15.3
19.2
17
tan x , x ⬇ 48
7
15.3
sin x , x ⬇ 27
Solve 䉭ABC by using the given measurements. Round measures of
sides to the nearest tenth and measures of angles to the nearest degree.
10. A ⫽ 35⬚, a ⫽ 12
b ⬇ 17.1, c ⬇ 20.9, B 55
11. B ⫽ 71⬚, b ⫽ 25
a ⬇ 8.6, c ⬇ 26.4, A 19
12. B ⫽ 36⬚, c ⫽ 8
13. a ⫽ 4, b ⫽ 7
14. A ⫽ 17⬚, c ⫽ 3.2
15. b ⫽ 52, c ⫽ 95
a ⬇ 6.5, b ⬇ 4.7, A 54
a ⬇ 0.9, b ⬇ 3.1, B 73
A
b
C
c
a
B
c ⬇ 8.1, A ⬇ 30, B ⬇ 60
a ⬇ 79.5, A ⬇ 33, B ⬇ 57
16. SURVEYING John stands 150 meters from a water tower and sights the top at an angle
© Glencoe/McGraw-Hill
778
NAME ______________________________________________ DATE
____________ PERIOD _____
Reading to Learn Mathematics
Right Triangle Trigonometry
Pre-Activity How is trigonometry used in building construction?
Read the introduction to Lesson 13-1 at the top of page 701 in your textbook.
If a different ramp is built so that the angle shown in the figure has a
1
14
tangent of ᎏᎏ, will this ramp meet, exceed, or fail to meet ADA regulations?
exceed
Reading the Lesson
r
1. Refer to the triangle at the right. Match each trigonometric
function with the correct ratio.
r
i. ᎏ
t
r
ii. ᎏ
s
t
iii. ᎏ
r
s
iv. ᎏ
t
s
v. ᎏ
r
␪
s
t
vi. ᎏ
s
a. sin ␪ iv
b. tan ␪ v
c. sec ␪ iii
d. cot ␪ ii
e. cos ␪ i
f. csc ␪ vi
t
2. Refer to the Key Concept box on page 703 in your textbook. Use the drawings of the
30⬚-60⬚-90⬚ triangle and 45⬚-45⬚-90⬚ triangle and/or the table to complete the following.
a. The tangent of 45⬚ and the
cotangent
b. The sine of 30⬚ is equal to the cosine of
c. The sine and
cosine
of 45⬚ are equal.
60
.
of 45⬚ are equal.
d. The reciprocal of the cosecant of 60⬚ is the
e. The reciprocal of the cosine of 30⬚ is the
f. The reciprocal of the tangent of 60⬚ is the
sine
cosecant
tangent
of 60⬚.
of 60⬚.
of 30⬚.
Helping You Remember
3. In studying trigonometry, it is important for you to know the relationships between the
lengths of the sides of a 30⬚-60⬚-90⬚ triangle. If you remember just one fact about this
triangle, you will always be able to figure out the lengths of all the sides. What fact can
you use, and why is it enough?
Sample answer: The shorter leg is half as long as the hypotenuse. You
can use the Pythagorean Theorem to find the length of the longer leg.
© Glencoe/McGraw-Hill
779
NAME ______________________________________________ DATE
____________ PERIOD _____
Enrichment
The Angle of Repose
Suppose you place a block of wood on an inclined
plane, as shown at the right. If the angle, ␪, at which
the plane is inclined from the horizontal is very small,
the block will not move. If you increase the angle, the
block will eventually overcome the force of friction and
start to slide down the plane.
For situations in which the block and plane are smooth
but unlubricated, the angle of repose depends only on
the types of materials in the block and the plane. The
angle is independent of the area of contact between
the two surfaces and of the weight of the block.
The drawing at the right shows how to use vectors to
find a coefficient of friction. This coefficient varies with
different materials and is denoted by the
Greek leter mu, ␮.
1. A wooden chute is built so that
wooden crates can slide down into the
basement of a store. What angle should
the chute make in order for the crates
to slide down at a constant speed?
d
line
Inc
ne
Pla
␪
At the instant the block begins to slide, the angle
formed by the plane is called the angle of friction, or
the angle of repose.
Solve each problem.
ck
Blo
F
␪
␪
F ⫽ W sin ␪
N ⫽ W cos
F ⫽ ␮N
sin ␪
ᎏ ⫽ tan ␪
␮⫽ᎏ
cos ␪
Material
Coefficient of Friction ␮
Wood on wood
Wood on stone
Rubber tire on dry concrete
Rubber tire on wet concrete
0.5
0.5
1.0
0.7
2. Will a 100-pound wooden crate slide down a stone ramp that makes an
angle of 20⬚ with the horizontal? Explain your answer.
3. If you increase the weight of the crate in Exercise 2 to 300 pounds, does it
change your answer?
4. A car with rubber tires is being driven on dry concrete pavement. If the
car tires spin without traction on a hill, how steep is the hill?
5. For Exercise 4, does it make a difference if it starts to rain? Explain your
answer.
© Glencoe/McGraw-Hill
N
W
␪
780
NAME ______________________________________________ DATE
____________ PERIOD _____
Study Guide and Intervention
Angles and Angle Measurement
Angle Measurement An angle is determined by two rays. The degree measure of an
angle is described by the amount and direction of rotation from the initial side along the
positive x-axis to the terminal side. A counterclockwise rotation is associated with positive
angle measure and a clockwise rotation is associated with negative angle measure. An angle
can also be measured in radians.
180⬚
Radian and
Degree
Measure
ᎏ
To rewrite the radian measure of an angle in degrees, multiply the number of radians by ᎏ
␲ radians .
␲ radᎏ
ians .
To rewrite the degree measure of an angle in radians, multiply the number of degrees by ᎏ
180⬚
Example 1
Example 2
Draw an angle with
measure 290 in standard notation.
The negative y-axis represents a positive
rotation of 270⬚. To generate an angle of
290⬚, rotate the terminal side 20⬚ more in
the counterclockwise direction.
Rewrite the degree
measure in radians and the radian
measure in degrees.
a. 45
冢 ␲ radians
180° 冣
␲
4
45⬚ ⫽ 45⬚ ᎏᎏ ⫽ ᎏ radians
y 90
5␲
3
b. radians
290
冢
冣
5␲
5␲ 180°
ᎏ radians ⫽ ᎏ ᎏ ⫽ 300⬚
3
3
␲
initial side x
O
180
terminal side
270
Exercises
Draw an angle with the given measure in standard position.
5␲
4
2. ⫺ ᎏ
1. 160⬚
3. 400⬚
y
O
y
x
y
x
O
O
x
Rewrite each degree measure in radians and each radian measure in degrees.
4. 140⬚
7
9
© Glencoe/McGraw-Hill
5. ⫺860⬚
43
9
3␲
5
6. ⫺ᎏᎏ
108
781
11␲
3
7. ᎏᎏ
660
NAME ______________________________________________ DATE
____________ PERIOD _____
Study Guide and Intervention (continued)
Angles and Angle Measurement
Coterminal Angles When two angles in standard position have the same terminal
sides, they are called coterminal angles. You can find an angle that is coterminal to a
given angle by adding or subtracting a multiple of 360⬚. In radian measure, a coterminal
angle is found by adding or subtracting a multiple of 2␲.
Example
Find one angle with positive measure and one angle with negative
measure coterminal with each angle.
a. 250
A positive angle is 250⬚ ⫹ 360⬚ or 610⬚.
A negative angle is 250⬚ ⫺ 360⬚ or ⫺110⬚.
5
8
b. 5␲
21␲
8
8
5␲
11␲
A negative angle is ᎏᎏ ⫺ 2␲ or ⫺ᎏᎏ.
8
8
A positive angle is ᎏᎏ ⫹ 2␲ or ᎏᎏ.
Exercises
Find one angle with a positive measure and one angle with a negative measure
coterminal with each angle. 1–18 Sample answers are given.
1. 65⬚
425, 295
4. 420⬚
60, 300
7. ⫺290⬚
70, 650
2. ⫺75⬚
285, 435
700, 20
330, 30
⫺7␲
4
14. ᎏᎏ
17␲
5
17. ᎏᎏ
19
17
, 9
9
13. ᎏᎏ
15
, 4
4
16. ᎏᎏ
7
3
, 5
5
© Glencoe/McGraw-Hill
230, 490
9. ⫺420⬚
8. 690⬚
11. ᎏᎏ
590, 130
6. ⫺130⬚
5. 340⬚
␲
9
10. ᎏᎏ
3. 230⬚
300, 60
3␲
8
12. ᎏᎏ
15␲
4
15. ᎏᎏ
⫺5␲
3
18. ᎏᎏ
19
13
, 8
8
6␲
5
16
4
, 5
5
⫺13␲
6
7
, 4
4
11
, 6
6
11
, 3
3
782
⫺11␲
4
5
3
, 4
4
NAME ______________________________________________ DATE
8-2
____________ PERIOD _____
Skills Practice
Angles and Angle Measure
Draw an angle with the given measure in standard position.
1. 185⬚
2. 810⬚
3. 390⬚
y
y
x
O
y
x
O
5. ⫺50⬚
4. 495⬚
6. ⫺420⬚
y
y
x
O
x
O
O
y
x
O
x
Rewrite each degree measure in radians and each radian measure in degrees.
13
18
8. 720⬚ 4
7
6
10. 90⬚ 7. 130⬚ 2
9. 210⬚ 6
3
2
11. ⫺30⬚ 12. ⫺270⬚ 13. ᎏᎏ 60
␲
3
14. ᎏᎏ 150
5␲
6
2␲
3
16. ᎏᎏ 225
5␲
4
15. ᎏᎏ 120
3␲
4
17. ⫺ᎏᎏ 135
7␲
6
18. ⫺ᎏᎏ 210
Find one angle with positive measure and one angle with negative measure
coterminal with each angle. 19–26. Sample answers are given.
19. 45⬚ 405, 315
20. 60⬚ 420, 300
21. 370⬚ 10, 350
22. ⫺90⬚ 270, 450
2␲ 8
3
3
4
3
24. ᎏᎏ , ␲ 13
6
6
11
6
26. ⫺ᎏᎏ , 23. ᎏᎏ , 25. ᎏᎏ , © Glencoe/McGraw-Hill
5␲ 9
2
2
3␲ 5
4
4
783
2
3
2
NAME ______________________________________________ DATE
8-2
____________ PERIOD _____
Practice (Average)
Angles and Angle Measure
Draw an angle with the given measure in standard position.
1. 210⬚
2. 305⬚
3. 580⬚
y
y
x
O
y
x
O
5. ⫺450⬚
4. 135⬚
y
6. ⫺560⬚
y
x
O
x
O
O
y
x
x
O
Rewrite each degree measure in radians and each radian measure in degrees.
10
30
7. 18⬚ 8. 6⬚ 2
5
41
9
11. ⫺72⬚ 12. ⫺820⬚ 15. 4␲ 720
16. ᎏᎏ 450
9␲
2
19. ⫺ᎏᎏ 810
29
6
25
13. ⫺250⬚ 18
9. 870⬚ 5␲
2
13␲
5
17. ᎏᎏ 468
7␲
12
20. ⫺ᎏᎏ 105
3␲
8
21. ⫺ᎏᎏ 67.5
347
180
11
14. ⫺165⬚ 12
10. 347⬚ 13␲
30
18. ᎏᎏ 78
3␲
16
22. ⫺ᎏᎏ 33.75
Find one angle with positive measure and one angle with negative measure
coterminal with each angle. 23–34. Sample answers are given.
23. 65⬚ 425, 295
24. 80⬚ 440, 280
25. 285⬚ 645, 75
26. 110⬚ 470, 250
27. ⫺37⬚ 323, 397
28. ⫺93⬚ 267, 453
2␲ 12
5
5
8
5
7
3␲ 32. ⫺ᎏᎏ , 2 2
2
29. ᎏᎏ , 5␲ 17
6
6
7
6
9
␲ 7
33. ⫺ᎏᎏ , 4 4
4
30. ᎏᎏ , 17␲ 29
6
6
7
6
29
5␲ 19
34. ⫺ᎏᎏ , 12 12
12
31. ᎏᎏ , 35. TIME Find both the degree and radian measures of the angle through which the hour
5
hand on a clock rotates from 5 A.M. to 10 A.M.
150; 6
36. ROTATION A truck with 16-inch radius wheels is driven at 77 feet per second
(52.5 miles per hour). Find the measure of the angle through which a point on the
outside of the wheel travels each second. Round to the nearest degree and nearest radian.
3309/s; 58 radians/s
© Glencoe/McGraw-Hill
784
NAME ______________________________________________ DATE
8-2
____________ PERIOD _____
Reading to Learn Mathematics
Angles and Angle Measure
Pre-Activity How can angles be used to describe circular motion?
Read the introduction to Lesson 13-2 at the top of page 709 in your textbook.
If a gondola revolves through a complete revolution in one minute, what is
its angular velocity in degrees per second? 6 per second
Reading the Lesson
1. Match each degree measure with the corresponding radian measure on the right.
2␲
3
a. 30⬚ v
i. ᎏᎏ
b. 90⬚ ii
ii. ᎏᎏ
c. 120⬚ i
iii. ᎏᎏ
d. 135⬚ vi
iv. ␲
e. 180⬚ iv
v. ᎏᎏ
f. 210⬚ iii
vi. ᎏᎏ
␲
2
7␲
6
␲
6
3␲
4
1
2
1
2
2. The sine of 30⬚ is ᎏᎏ and the sine of 150⬚ is also ᎏᎏ. Does this mean that 30⬚ and 150⬚ are
coterminal angles? Explain your reasoning. Sample answer: No; the terminal
side of a 30 angle is in Quadrant I, while the terminal side of a 150 angle
is in Quadrant II.
3. Describe how to find two angles that are coterminal with an angle of 155⬚, one with
positive measure and one with negative measure. (Do not actually calculate these angles.)
Sample answer: Positive angle: Add 360 to 155. Negative angle:
Subtract 360 from 155.
5␲
3
4. Describe how to find two angles that are coterminal with an angle of ᎏᎏ, one positive and
one negative. (Do not actually calculate these angles.) Sample answer: Positive
5
3
5
3
angle: Add 2 to . Negative angle: Subtract 2 from .
Helping You Remember
5. How can you use what you know about the circumference of a circle to remember how to
convert between radian and degree measure? Sample answer: The
circumference of a circle is given by the formula C 2r, so the
circumference of a circle with radius 1 is 2. In degree measure, one
complete circle is 360. So 2 radians 360 and radians 180.
© Glencoe/McGraw-Hill
785
NAME ______________________________________________ DATE
8-2
____________ PERIOD _____
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.
w
stra
10 cm
Your eye
7 cm
weight
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 free-hanging
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
© Glencoe/McGraw-Hill
786
NAME ______________________________________________ DATE
8-3
____________ PERIOD _____
Study Guide and Intervention
Trigonometric Functions of General Angles
Trigonometric Functions and General Angles
Trigonometric Functions,
␪ in Standard Position
Let ␪ be an angle in standard position and let P (x, y ) be a point on the terminal side
of ␪. By the Pythagorean Theorem, the distance r from the origin is given by
y
x 2 ⫹ y 2苶. The trigonometric functions of an angle in standard position may be
r ⫽ 兹苶
defined as follows.
P(x, y )
r
␪
y
y
r
cos ␪ ⫽ ᎏ
x
r
tan ␪ ⫽ ᎏ
r
y
sec ␪ ⫽ ᎏ
r
x
cot ␪ ⫽ ᎏ
csc ␪ ⫽ ᎏ
x
O
y
x
sin ␪ ⫽ ᎏ
x
x
y
Example
Find the exact values of the six trigonometric functions of ␪ if the
苶 ).
terminal side of ␪ contains the point (5, 5兹2
2. You need to find r.
You know that x ⫽ ⫺5 and y ⫽ 5兹苶
r ⫽ 兹苶
x2 ⫹ y2苶
⫽ 兹苶苶
(⫺5)2 ⫹ (5兹2
苶) 2
Pythagorean Theorem
Replace x with ⫺5 and y with 5兹2
苶.
⫽ 兹75
苶 or 5兹3
苶
Now use x ⫽ ⫺5, y ⫽ 5兹2
苶, and r ⫽ 5兹3
苶 to write the ratios.
5兹2
苶
兹6
苶
5兹3
苶
5兹3
苶
兹6
苶
r
csc ␪ ⫽ ᎏᎏ ⫽ ᎏ ⫽ ᎏ
2
y
5兹2
苶
y
r
sin ␪ ⫽ ᎏᎏ ⫽ ᎏ ⫽ ᎏ
3
兹3
苶
⫺5
5兹3
苶
5兹3
苶
r
sec ␪ ⫽ ᎏᎏ ⫽ ᎏ
⫽ ⫺兹3
苶
⫺5
x
x
r
y
5兹2
苶
x
y
⫺5
5兹2
苶
tan ␪ ⫽ ᎏxᎏ ⫽ ᎏ
⫽ ⫺兹2
苶
⫺5
cos ␪ ⫽ ᎏᎏ ⫽ ᎏ ⫽ ⫺ ᎏ
3
兹2
苶
cot ␪ ⫽ ᎏᎏ ⫽ ᎏ ⫽ ⫺ ᎏ
2
Exercises
Find the exact values of the six trigonometric functions of ␪ if the terminal side of
␪ in standard position contains the given point.
2. (4, 4兹3
苶)
1. (8, 4)
1
2兹5
苶
2
5
5
5
兹苶
苶, sec ␪ , cot ␪ 2
csc ␪ 兹5
2
兹5
苶
3. (0, ⫺4)
3
兹苶
1
2
兹1
苶0
3兹苶
10
10
sin ␪ , cos ␪ , tan ␪ 兹3
苶,
sin ␪ , cos ␪ , tan ␪ ,
2
2兹3
苶
兹3
苶
csc ␪ , sec ␪ 2, cot ␪ 3
3
4. (6, 2)
sin ␪ 1, cos ␪ 0,
sin ␪ , cos ␪ , tan ␪ tan ␪ undefined, csc ␪ 1,
1
兹10
苶
, csc ␪ 兹10
苶
, sec ␪ ,
3
3
10
sec ␪ undefined , cot ␪ 0
© Glencoe/McGraw-Hill
cot ␪ 3
787
NAME ______________________________________________ DATE
8-3
____________ PERIOD _____
Study Guide and Intervention (continued)
Trigonometric Functions of General Angles
Reference Angles If ␪ is a nonquadrantal angle in standard position, its reference
angle ␪⬘ is defined as the acute angle formed by the terminal side of ␪ and the x-axis.
Reference
Angle Rule
y Quadrant I
Quadrant II y
␪
␪
␪
␪
x
O
␪
y
␪
O x
O
␪
x
x
O
Quadrant III
␪ ␪
y
␪ 180 ␪
(␪ ␪)
Quadrant IV
␪ ␪ 180
(␪ ␪ )
␪ 360 ␪
(␪ 2 ␪)
Quadrant
Signs of
Trigonometric
Functions
Function
I
II
III
IV
sin ␪ or csc ␪
⫹
⫹
⫺
⫺
cos␪ or sec ␪
⫹
⫺
⫺
⫹
tan ␪ or cot ␪
⫹
⫺
⫹
⫺
Example 1
Sketch an angle of measure
205. Then find its reference angle.
Because the terminal side of 205° lies in
Quadrant III, the reference angle ␪⬘ is
205⬚ ⫺ 180⬚ or 25⬚.
y
Use a reference angle
3
4
to find the exact value of cos .
3␲
4
Because the terminal side of ᎏ lies in
Quadrant II, the reference angle ␪⬘ is
3␲
4
␲
4
␲ ⫺ ᎏ or ᎏ .
␪ 205
The cosine function is negative in
Quadrant II.
x
O
␪
Example 2
3␲
4
␲
4
兹2
苶
cos ᎏ ⫽ ⫺cos ᎏ ⫽ ⫺ ᎏ
.
2
Exercises
Find the exact value of each trigonometric function.
3
兹苶
1. tan(⫺510⬚) 3
2. csc ᎏ 兹2
苶
3. sin(⫺90⬚) 1
4. cot 1665⬚ 1
5. cot 30⬚ 兹3
苶
6. tan 315⬚ 1
␲
4
7. csc ᎏ 兹2
苶
© Glencoe/McGraw-Hill
11␲
4
4␲
3
8. tan ᎏ
788
NAME ______________________________________________ DATE
8-3
____________ PERIOD _____
Skills Practice
Trigonometric Functions of General Angles
Find the exact values of the six trigonometric functions of ␪ if the terminal side of
␪ in standard position contains the given point.
1. (5, 12)
2. (3, 4)
12
12
5
sin ␪ , cos ␪ , tan ␪ ,
13
5
13
13
13
5
csc ␪ , sec ␪ , cot ␪ 12
5
12
4
3
4
5
5
3
5
5
3
csc ␪ , sec ␪ , cot ␪ 4
3
4
sin ␪ , cos ␪ , tan ␪ ,
3. (8, ⫺15)
4. (⫺4, 3)
8
15
15
sin ␪ , cos ␪ , tan ␪ ,
17
17
8
8
17
17
csc ␪ , sec ␪ , cot ␪ 15
15
8
5. (⫺9, ⫺40)
3
4
3
5
5
4
5
5
4
csc ␪ , sec ␪ , cot ␪ 3
4
3
sin ␪ , cos ␪ , tan ␪ ,
6. (1, 2)
9
40
40
sin ␪ , cos ␪ , tan ␪ ,
41
41
9
2兹苶5
5
2,
41
40
9
40
41
9
兹5
苶
sin ␪ , cos ␪ , tan ␪ 5
1
2
5
兹苶
csc ␪ , sec ␪ 兹5
苶, cot ␪ csc ␪ , sec ␪ , cot ␪ 2
Sketch each angle. Then find its reference angle.
7. 135⬚ 45
8. 200⬚ 20
y
y
x
O
5␲ 3 3
9. ᎏᎏ y
x
O
O
x
Find the exact value of each trigonometric function.
1
2
10. sin 150⬚ 11. cos 270⬚ 0
␲
4
15. cos ᎏᎏ 14. tan ᎏᎏ 1
4␲
3
12. cot 135⬚ 1
3
兹苶
13. tan (⫺30⬚) 3
2
兹苶
3␲
16. cot (⫺␲)
17. sin 冢⫺ᎏᎏ冣 4
2
undefined
1
2
Suppose ␪ is an angle in standard position whose terminal side is in the given
quadrant. For each function, find the exact values of the remaining five
trigonometric functions of ␪.
4
5
12
5
19. tan ␪ ⫽ ⫺ᎏᎏ, Quadrant IV
18. sin ␪ ⫽ ᎏᎏ, Quadrant II
3
4
5
3
5
3
sec ␪ , cot ␪ 3
4
5
4
cos ␪ , tan ␪ , csc ␪ ,
© Glencoe/McGraw-Hill
12
5
13
13
13
5
sec ␪ , cot ␪ 5
12
13
12
sin ␪ , cos ␪ , csc ␪ ,
789
NAME ______________________________________________ DATE
8-3
____________ PERIOD _____
Practice (Average)
Trigonometric Functions of General Angles
Find the exact values of the six trigonometric functions of ␪ if the terminal side of
␪ in standard position contains the given point.
29
5兹苶
3. (⫺2, ⫺5) sin ␪ ,
1. (6, 8)
2. (⫺20, 21)
4
3
sin ␪ , cos ␪ ,
5
5
4
5
tan ␪ , csc ␪ ,
3
4
5
3
sec ␪ , cot ␪ 3
4
21
20
sin ␪ , cos ␪ ,
29
29
21
29
tan ␪ , csc ␪ ,
20
21
29
20
sec ␪ , cot ␪ 20
21
29
29
2兹苶
cos ␪ ,
29
5
兹29
苶
tan ␪ , csc ␪ ,
2
5
2
兹29
苶
sec ␪ , cot ␪ 5
2
Find the reference angle for the angle with the given measure.
13␲ 3
8
8
4. 236⬚ 56
7␲ 4 4
6. ⫺210⬚ 30
5. ᎏᎏ 7. ⫺ᎏᎏ Find the exact value of each trigonometric function.
8. tan 135⬚ 1
5␲
3
12. tan ᎏᎏ 兹3
苶
9. cot 210⬚ 兹3
苶
冢 34␲ 冣
13. csc ⫺ᎏᎏ 兹2
苶
10. cot (⫺90⬚) 0
11. cos 405⬚ 14. cot 2␲
苶
13␲ 兹3
15. tan ᎏᎏ undefined
2
3
6
Suppose ␪ is an angle in standard position whose terminal side is in the given
quadrant. For each function, find the exact values of the remaining five
trigonometric functions of ␪.
兹5
苶
2
17. sin ␪ ⫽ ᎏᎏ, Quadrant III cos ␪ ,
12
5
16. tan ␪ ⫽ ⫺ᎏᎏ, Quadrant IV
3
2兹苶5
tan ␪ , csc ␪ ,
2
5
3兹5
5
苶
兹苶
sec ␪ , cot ␪ 5
2
18. LIGHT Light rays that “bounce off” a surface are reflected
by the surface. If the surface is partially transparent, some
of the light rays are bent or refracted as they pass from the
air through the material. The angles of reflection ␪1 and of
refraction ␪2 in the diagram at the right are related by the
3, find the
equation sin ␪1 ⫽ n sin ␪2. If ␪1 ⫽ 60⬚ and n ⫽ 兹苶
measure of ␪2. 30
air
␪1
␪2
800 N
800兹苵
3N
; 60
2
␪
790
␪1
surface
19. FORCE A cable running from the top of a utility pole to the
ground exerts a horizontal pull of 800 Newtons and a vertical
3 Newtons. What is the sine of the angle ␪ between the
pull of 800兹苶
3
cable and the ground? What is the measure of this angle? 兹苶
© Glencoe/McGraw-Hill
3
3
5
12
13
sin ␪ , cos ␪ , csc ␪ ,
13
13
12
13
5
sec ␪ , cot ␪ 5
12
NAME ______________________________________________ DATE
8-3
____________ PERIOD _____
Reading to Learn Mathematics
Trigonometric Functions of General Angles
Pre-Activity How can you model the position of riders on a skycoaster?
Read the introduction to Lesson 13-3 at the top of page 717 in your textbook.
• What does t ⫽ 0 represent in this application? Sample answer: the
time when the riders leave the bottom of their swing
• Do negative values of t make sense in this application? Explain your
answer. Sample answer: No; t 0 represents the starting
time, so the value of t cannot be less than 0.
Reading the Lesson
1. Suppose ␪ is an angle in standard position, P(x, y) is a point on the terminal side of ␪,
and the distance from the origin to P is r. Determine whether each of the following
statements is true or false.
a. The value of r can be found by using either the Pythagorean Theorem or the distance
formula. true
x
r
b. cos ␪ ⫽ ᎏ true
c. csc ␪ is defined if y ⫽ 0. true
d. tan ␪ is undefined if y ⫽ 0. false
e. sin ␪ is defined for every value of ␪. true
2. Let ␪ be an angle measured in degrees. Match the quadrant of ␪ from the first column
with the description of how to find the reference angle for ␪ from the second column.
a. Quadrant III ii
i. Subtract ␪ from 360⬚.
b. Quadrant IV i
ii. Subtract 180⬚ from ␪.
c. Quadrant II iv
iii. ␪ is its own reference angle.
d. Quadrant I iii
iv. Subtract ␪ from 180⬚.
Helping You Remember
3. The chart on page 719 in your textbook summarizes the signs of the six trigonometric
functions in the four quadrants. Since reciprocals always have the same sign, you only
need to remember where the sine, cosine, and tangent are positive. How can you
remember this with a simple diagram?
Sample answer:
y
O
© Glencoe/McGraw-Hill
x
791
NAME ______________________________________________ DATE
8-3
____________ PERIOD _____
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.
Area of circle
AC ⫽ ␲r 2
Area of inscribed polygon
AI ⫽ ᎏᎏ ⫻ sin ᎏᎏ
Area of circumscribed polygon
AC ⫽ nr2 ⫻ tan ᎏᎏ
nr2
2
360°
n
r
r
180°
n
Use a calculator to complete the chart below for a unit circle
(a circle of radius 1).
Number
of Sides
3
1.
4
2.
8
3.
12
4.
20
5.
24
6.
28
7.
32
8.
1000
Area of
Inscribed
Polygon
Area of Circle
minus
Area of Polygon
Area of
Circumscribed
Polygon
Area of Polygon
minus
Area of Circle
1.2990381
1.8425545
5.1961524
2.054597
9. What number do the areas of the circumscribed and inscribed polygons
seem to be approaching?
© Glencoe/McGraw-Hill
792
NAME ______________________________________________ DATE
8-4
____________ PERIOD _____
Study Guide and Intervention
Inverse Trigonometric Functions
Solve Equations Using Inverses If the domains of trigonometric functions are
restricted to their principal values, then their inverses are also functions.
Principal Values
of Sine, Cosine,
and Tangent
y ⫽ Sin x if and only if y ⫽ sin x and ⫺ᎏ␲ᎏ ⱕ x ⱕ ᎏ␲ᎏ.
2
2
␲
␲
y ⫽ Cos x if and only if y ⫽ cos x and 0 ⱕ x ⱕ ␲.
y ⫽ Tan x if and only if y ⫽ tan x and ⫺ᎏ2ᎏ ⱕ x ⱕ ᎏ2ᎏ.
Inverse Sine,
Cosine, and
Tangent
Given y ⫽ Sin x, the inverse Sine function is defined by y ⫽ Sin⫺1 x or y ⫽ Arcsin x.
Given y ⫽ Cos x, the inverse Cosine function is defined by y ⫽ Cos⫺1 x or y ⫽ Arccos x.
Given y ⫽ Tan x, the inverse Tangent function is given by y ⫽ Tan⫺1 x or y ⫽ Arctan x.
Example 1
Solve x ⫽ Sin⫺1 ᎏᎏ .
冢 兹3苶 冣
冢 兹23苶 冣
兹3
苶
␲
2
␲
2
, then Sin x ⫽ ᎏ
and ⫺ᎏᎏ ⱕ x ⱕ ᎏᎏ.
If x ⫽ Sin⫺1 ᎏ
2
2
␲
3
The only x that satisfies both criteria is x ⫽ ᎏᎏ or 60⬚.
Example 2
冢 兹3苶3 冣
Solve Arctan ⫺ᎏᎏ ⫽ x.
冢 兹3苶 冣
兹3
苶
␲
␲
, then Tan x ⫽ ⫺ ᎏ
and ⫺ᎏᎏ ⱕ x ⱕ ᎏᎏ.
If x ⫽ Arctan ⫺ ᎏ
3
3
2
2
␲
6
The only x that satisfies both criteria is ⫺ᎏᎏ or ⫺30⬚.
Exercises
Solve each equation by finding the value of x to the nearest degree.
1. Cos⫺1 ⫺ ᎏ
⫽ x 150⬚
2
冢 兹3苶 冣
2. x ⫽ Sin⫺1 ᎏ
60⬚
2
3. x ⫽ Arccos (⫺0.8) 143⬚
4. x ⫽ Arctan 兹3
苶 60⬚
冢 兹2苶 冣
兹3
苶
5. x ⫽ Arccos ⫺ ᎏ
135⬚
2
6. x ⫽ Tan⫺1 (⫺1) ⫺45⬚
7. Sin⫺1 0.45 ⫽ x 27⬚
8. x ⫽ Arcsin ⫺ ᎏ
⫺60⬚
2
冢 12 冣
冢 兹3苶 冣
9. x ⫽ Arccos ⫺ᎏᎏ 120⬚
10. Cos⫺1 (⫺0.2) ⫽ x 102⬚
11. x ⫽ Tan⫺1 (⫺兹3
苶) ⫺60⬚
12. x ⫽ Arcsin 0.3 17⬚
13. x ⫽ Tan⫺1 (15) 86⬚
14. x ⫽ Cos⫺1 1 0
15. Arctan⫺1 (⫺3) ⫽ x ⫺72⬚
16. x ⫽ Sin⫺1 (⫺0.9) ⫺64⬚
17. Arccos⫺1 0.15 81⬚
18. x ⫽ Tan⫺1 0.2 11⬚
© Glencoe/McGraw-Hill
811
NAME ______________________________________________ DATE
8-4
____________ PERIOD _____
Study Guide and Intervention (continued)
Inverse Trigonometric Functions
Trigonometric Values You can use a calculator to find the values of trigonometric
expressions.
Example
Find each value. Write angle measures in radians. Round to the
nearest hundredth.
冢
1
2
冣
a. Find tan Sin⫺1 ᎏᎏ .
1
1
␲
␲
2
2
2
2
兹3
苶
兹3
苶
␲
1
⫺1
so tan Sin ᎏᎏ ⫽ ᎏ
.
conditions. tan ᎏᎏ ⫽ ᎏ
3
3
6
2
␲
6
Let ␪ ⫽ Sin⫺1 ᎏᎏ. Then Sin ␪ ⫽ ᎏᎏ with ⫺ᎏᎏ ⬍ ␪ ⬍ ᎏᎏ. The value ␪ ⫽ ᎏᎏ satisfies both
冢
冣
b. Find cos (Tan⫺1 4.2).
KEYSTROKES: COS 2nd [tan–1] 4.2 ENTER .2316205273
Therefore cos (Tan⫺1 4.2) ⬇ 0.23.
Exercises
Find each value. Write angle measures in radians. Round to the nearest
hundredth.
1
2
1. cot (Tan⫺1 2) ᎏᎏ
2. Arctan(⫺1) ⫺0.79
冤
冢 兹2苶 冣冥 0.71
冤
冢 57 冣冥 ⫺1.02 8. sin 冢Tan⫺1 ᎏ15ᎏ2 冣 0.38
4. cos Sin⫺1 ⫺ ᎏ
2
冢 兹3苶 冣
5. Sin⫺1 ⫺ ᎏ
⫺1.05
2
7. tan Arcsin ⫺ᎏᎏ
冢 兹3苶 冣
3. cot⫺1 1 1.27
冢
兹3
苶
冣
6. sin Arcsin ᎏ
0.87
2
9. sin [Arctan⫺1 (⫺兹2
苶)] ⫺0.82
10. Arccos ⫺ ᎏ
2.62
2
11. Arcsin ᎏ
1.05
2
冢 兹3苶 冣
12. Arccot ⫺ ᎏ
⫺1.91
3
13. cos [Arcsin (⫺0.7)] 0.71
14. tan (Cos⫺1 0.28) 3.43
15. cos (Arctan 5) 0.20
16. Sin⫺1 (⫺0.78) ⫺0.89
17. Cos⫺1 0.42 1.14
18. Arctan (⫺0.42) ⫺0.40
19. sin (Cos⫺1 0.32) 0.95
20. cos (Arctan 8) 0.12
21. tan (Cos⫺1 0.95) 0.33
© Glencoe/McGraw-Hill
812
冢 兹3苶 冣
NAME ______________________________________________ DATE
8-4
____________ PERIOD _____
Skills Practice
Inverse Trigonometric Functions
Write each equation in the form of an inverse function.
1. ␣ ⫽ cos ␤ ␤ ⫽ cos⫺1 ␣
2. sin b ⫽ a sin⫺1 a ⫽ b
3. y ⫽ tan x x ⫽ tan⫺1 y
2
兹苶
兹2
苶
4. cos 45⬚ ⫽ ᎏ
cos⫺1 ᎏᎏ ⫽ 45⬚
2
5. b ⫽ sin 150⬚ 150⬚ ⫽ sin⫺1 b
6. tan y ⫽ ᎏᎏ tan⫺1 ᎏᎏ ⫽ y
2
4
5
4
5
Solve each equation by finding the value of x to the nearest degree.
7. x ⫽ Cos⫺1 (⫺1) 180⬚
8. Sin⫺1 (⫺1) ⫽ x ⫺90⬚
9. Tan⫺1 1 ⫽ x 45⬚
10. x ⫽ Arcsin ⫺ ᎏ
⫺60⬚
2
11. x ⫽ Arctan 0 0⬚
12. x ⫽ Arccos ᎏᎏ 60⬚
冢 兹3苶 冣
1
2
Find each value. Write angle measures in radians. Round to the nearest
hundredth.
兹2
苶
冢 兹3苶 冣
0.79 radians
13. Sin⫺1 ᎏ
2
14. Cos⫺1 ⫺ ᎏ
2.62 radians
2
15. Tan⫺1 兹3
苶 1.05 radians
16. Arctan ⫺ ᎏ
⫺0.52 radians
3
冢 兹2苶 冣
冢 兹3苶 冣
17. Arccos ⫺ ᎏ
2.36 radians
2
18. Arcsin 1 1.57 radians
19. sin (Cos⫺1 1) 0
20. sin Sin⫺1 ᎏᎏ 0.5
冢
兹3
苶
冢
冣
1
2
冣
21. tan Arcsin ᎏ
1.73
2
22. cos (Tan⫺1 3) 0.32
23. sin [Arctan (⫺1)] ⫺0.71
24. sin Arccos ⫺ ᎏ
2
© Glencoe/McGraw-Hill
冤
813
冢 兹2苶 冣冥 0.71
NAME ______________________________________________ DATE
8-4
____________ PERIOD _____
Practice (Average)
Inverse Trigonometric Functions
Write each equation in the form of an inverse function.
1. ␤ ⫽ cos ␣
2. tan ␤ ⫽ ␣
␣ ⫽ cos⫺1 ␤
␤ ⫽ tan⫺1 ␣
120⬚ ⫽ tan⫺1 y
兹3
苶
2␲
3
1
2
4. ⫺ᎏᎏ ⫽ cos x
3. y ⫽ tan 120⬚
5. sin ᎏᎏ ⫽ ᎏ
2
冢 12 冣
x ⫽ cos⫺1 ⫺ᎏᎏ
2␲
3
3
兹苶
sin⫺1 ᎏᎏ ⫽ ᎏᎏ
2
␲
3
1
2
6. cos ᎏᎏ ⫽ ᎏᎏ
␲
3
1
2
cos⫺1 ᎏᎏ ⫽ ᎏᎏ
Solve each equation by finding the value of x to the nearest degree.
7. Arcsin 1 ⫽ x 90⬚
兹2
苶
10. x ⫽ Arccos ᎏ
45⬚
2
兹3
苶
冢 兹3苶 冣
8. Cos⫺1 ᎏ
⫽ x 30⬚
2
9. x ⫽ tan⫺1 ⫺ ᎏ
⫺30⬚
3
11. x ⫽ Arctan (⫺兹3
苶 ) ⫺60⬚
12. Sin⫺1 ⫺ᎏᎏ ⫽ x ⫺30⬚
冢 12 冣
Find each value. Write angle measures in radians. Round to the nearest
hundredth.
冢 兹3苶 冣
13. Cos⫺1 ⫺ ᎏ
2
2.62 radians
冢
1
2
冣
16. tan Cos⫺1 ᎏᎏ
1.73
冢 兹2苶 冣
14. Sin⫺1 ⫺ ᎏ
2
⫺0.79 radians
冤
冢 35 冣冥
17. cos Sin⫺1 ⫺ᎏᎏ
0.8
冢
12
13
19. tan sin⫺1 ᎏᎏ
冣
2.4
冢
␲
3
冣
0.52 radians
⫺0.52 radians
18. cos [Arctan (⫺1)]
0.71
冢
兹3
苶
20. sin Arctan ᎏ
3
冣
0.5
22. Sin⫺1 cos ᎏᎏ
冢 兹3苶 冣
15. Arctan ⫺ ᎏ
3
冢
3␲
4
冣
21. Cos⫺1 tan ᎏᎏ
3.14 radians
冢
15
17
23. sin 2 Cos⫺1 ᎏᎏ
冣
冢
⫺0.5
0.83
兹3
苶
24. cos 2 Sin⫺1 ᎏ
2
冣
25. PULLEYS The equation x ⫽ cos⫺1 0.95 describes the angle through which pulley A moves,
and y ⫽ cos⫺1 0.17 describes the angle through which pulley B moves. Both angles are
greater than 270⬚ and less than 360⬚. Which pulley moves through a greater angle?
pulley A
26. FLYWHEELS The equation y ⫽ Arctan 1 describes the counterclockwise angle through
which a flywheel rotates in 1 millisecond. Through how many degrees has the flywheel
rotated after 25 milliseconds? 1125⬚
© Glencoe/McGraw-Hill
814
NAME ______________________________________________ DATE
8-4
____________ PERIOD _____
Reading to Learn Mathematics
Inverse Trigonometric Functions
Pre-Activity How are inverse trigonometric functions used in road design?
Read the introduction to Lesson 13-7 at the top of page 746 in your
textbook.
Suppose you are given specific values for v and r. What feature of your
graphing calculator could you use to find the approximate measure of the
banking angle ␪ ? Sample answer: the TABLE feature
Reading the Lesson
1. Indicate whether each statement is true or false.
a. The domain of the function y ⫽ sin x is the set of all real numbers. true
b. The domain of the function y ⫽ Cos x is 0 ⱕ x ⱕ ␲. true
c. The range of the function y ⫽ Tan x is ⫺1 ⱕ y ⱕ 1. false
␲
2
␲
2
d. The domain of the function y ⫽ Cos⫺1 x is ⫺ᎏᎏ ⱕ x ⱕ ᎏᎏ. false
e. The domain of the function y ⫽ Tan⫺1 x is the set of all real numbers. true
f. The range of the function y ⫽ Arcsin x is 0 ⱕ x ⱕ ␲. false
2. Answer each question in your own words.
a. What is the difference between the functions y ⫽ sin x and the function y ⫽ Sin x?
Sample answer: The domain of y ⫽ sin x is the set of all real numbers,
␲
␲
while the domain of y ⫽ Sin x is restricted to ⫺ᎏᎏ ⱕ x ⱕ ᎏᎏ.
2
2
b. Why is it necessary to restrict the domains of the trigonometric functions in order to
define their inverses? Sample answer: Only one-to-one functions have
inverses. None of the six basic trigonometric functions is one-to-one,
but related one-to-one functions can be formed if the domains are
restricted in certain ways.
Helping You Remember
3. What is a good way to remember the domains of the functions
y ⫽ Sin x, y ⫽ Cos x, and y ⫽ Tan x, which are also the range
of the functions y ⫽ Arcsin x, y ⫽ Arccos x, and y ⫽ Arctan x?
(You may want to draw a diagram.) Sample answer: Each
restricted domain must include an interval of
numbers for which the function values are positive
and one for which they are negative.
© Glencoe/McGraw-Hill
815
NAME ______________________________________________ DATE
8-4
____________ PERIOD _____
Enrichment
Snell’s Law
Snell’s Law describes what happens to a ray of light that passes from air into
water or some other substance. In the figure, the ray starts at the left and
makes an angle of incidence ␪ with the surface.
Part of the ray is reflected, creating an angle of reflection ␪. The rest of the
ray is bent, or refracted, as it passes through the other medium. This creates
angle ␪⬘.
The angle of incidence equals the angle of reflection.
The angles of incidence and refraction are related by Snell’s Law:
sin ␪ ⫽ k sin ␪⬘
The constant k is called the index of refraction.
␪
␪
k
Substance
1.33
Water
1.36
Ethyl alcohol
1.54
Rock salt and Quartz
1.46–1.96
␪'
2.42
Glass
Diamond
Use Snell’s Law to solve the following. Round angle measures to the
nearest tenth of a degree.
1. If the angle of incidence at which a ray of light strikes the surface of a
window is 45⬚ and k ⫽ 1.6, what is the measure of the angle of refraction?
2. If the angle of incidence of a ray of light that strikes the surface of water
is 50⬚, what is the angle of refraction?
3. If the angle of refraction of a ray of light striking a quartz crystal is 24⬚,
what is the angle of incidence?
4. The angles of incidence and refraction for rays of light were measured five
times for a certain substance. The measurements (one of which was in
error) are shown in the table. Was the substance glass, quartz, or diamond?
␪
15⬚
30⬚
40⬚
60⬚
80⬚
␪⬘
9.7⬚
16.1⬚
21.2⬚
28.6⬚
33.2⬚
5. If the angle of incidence at which a ray of light strikes the surface of
ethyl alcohol is 60⬚, what is the angle of refraction?
© Glencoe/McGraw-Hill
816
NAME ______________________________________________ DATE
8-5
____________ PERIOD _____
Study Guide and Intervention
Graphing Trigonometric Functions
Graph Trigonometric Functions To graph a trigonometric function, make a table of
values for known degree measures (0, 30, 45, 60, 90, and so on). Round function values to
the nearest tenth, and plot the points. Then connect the points with a smooth, continuous
curve. The period of the sine, cosine, secant, and cosecant functions is 360 or 2 radians.
The amplitude of the graph of a periodic function is the absolute value of half the
difference between its maximum and minimum values.
Amplitude of a Function
Example
Graph y ⫽ sin ␪ for ⫺360⬚ ⱕ ␪ ⱕ 0⬚.
First make a table of values.
␪
360°
330°
315°
300°
270°
240°
225°
210°
180°
sin ␪
0
1
2
2
兹苶
2
3
兹苶
2
1
3
兹苶
2
2
兹苶
2
1
2
0
␪
150°
135°
120°
90°
60°
45°
30°
0°
sin ␪
1
2
兹2
苶
兹3
苶
1
兹3
苶
兹2
苶
1
2
0
sin
1.0
2
2
2
2
y
y
0.5
360
270
180
O
90
0.5
1.0
Exercises
Graph the following functions for the given domain.
1. cos ␪, 360 ␪ 0
2. tan ␪, 2 ␪ 0
y
y
4
1
2
360
270
180
90
O
2
O
3
2
2
1
2
4
What is the amplitude of each function?
3.
4.
y
O
y
x
2
O
© Glencoe/McGraw-Hill
837
2
x
NAME ______________________________________________ DATE
8-5
____________ PERIOD _____
Study Guide and Intervention (continued)
Graphing Trigonometric Functions
Variations of Trigonometric Functions
For functions of the form y a sin b␪ and y a cos b␪, the amplitude is | a |,
360°
|b |
2
|b |
and the period is or .
Amplitudes
and Periods
For functions of the form y a tan b␪, the amplitude is not defined,
180°
|b |
|b |
and the period is or .
Example
Find the amplitude and period of each function. Then graph the
function.
␪
3
1
2
a. y ⫽ 4 cos ᎏᎏ
b. y ⫽ ⫺ᎏᎏ tan 2␪
The amplitude is not defined, and the
period is .
First, find the amplitude.
| a | | 4 |, so the amplitude is 4.
Next find the period.
2
360°
1080
4
冨 冨
1
3
y
2
Use the amplitude and period to help
graph the function.
O
4
y
4
y
4 cos –3
3
4
–4
2
O
2
–2
180
360
540
720
900 1080
2
4
Exercises
Find the amplitude, if it exists, and period of each function. Then graph each
function.
␪
2
1. y 3 sin ␪
2. y 2 tan y
y
2
2
O
O
90⬚ 180⬚ 270⬚ 360⬚
⫺2
⫺2
© Glencoe/McGraw-Hill
838
␲
2
␲
3␲
2
2␲
5␲
2
3␲
NAME ______________________________________________ DATE
8-5
____________ PERIOD _____
Skills Practice
Graphing Trigonometric Functions
Find the amplitude, if it exists, and period of each function. Then graph each
function.
1. y 2 cos ␪
2. y 4 sin ␪
y
3. y 2 sec ␪
y
y
2
4
4
1
2
2
O
90⬚ 180⬚ 270⬚ 360⬚
O
90⬚ 180⬚ 270⬚ 360⬚
O
⫺1
⫺2
⫺2
⫺2
⫺4
⫺4
1
2
4. y tan ␪
5. y sin 3␪
y
6. y csc 3␪
y
y
2
2
4
1
1
2
O
90⬚ 180⬚ 270⬚ 360⬚
O
90⬚ 180⬚ 270⬚ 360⬚
O
⫺1
⫺1
⫺2
⫺2
⫺2
⫺4
7. y tan 2␪
y
4
2
1
2
90⬚ 135⬚ 180⬚
O
45⬚
90⬚ 135⬚ 180⬚
O
⫺2
⫺1
⫺2
⫺4
⫺2
⫺4
© Glencoe/McGraw-Hill
150⬚
y
2
45⬚
90⬚
9. y 4 sin ␪
4
O
30⬚
1
2
8. y cos 2␪
y
90⬚ 180⬚ 270⬚ 360⬚
839
180⬚ 360⬚ 540⬚ 720⬚
NAME ______________________________________________ DATE
8-5
____________ PERIOD _____
Practice
Graphing Trigonometric Functions
Find the amplitude, if it exists, and period of each function. Then graph each
function.
1
2
2. y cot ␪
1. y 4 sin ␪
y
y
y
4
4
2
2
O
3. y cos 5␪
90
180
270
O
360
2
2
4
4
3
4
1
90
180
O
360
45
90
135
180
1
1
2
4. y csc ␪
270
5. y 2 tan ␪
6. 2y sin ␪
FORCE For Exercises 7 and 8, use the following information.
An anchoring cable exerts a force of 500 Newtons on a pole. The force has
the horizontal and vertical components Fx and Fy. (A force of one Newton (N),
is the force that gives an acceleration of 1 m/sec2 to a mass of 1 kg.)
7. The function Fx 500 cos ␪ describes the relationship between the
angle ␪ and the horizontal force. What are the amplitude and period
of this function?
500 N
Fy
Fx
8. The function Fy 500 sin ␪ describes the relationship between the angle ␪ and the
vertical force. What are the amplitude and period of this function?
WEATHER For Exercises 9 and 10, use the following information.
The function y 60 25 sin t, where t is in months and t 0 corresponds to April 15,
6
models the average high temperature in degrees Fahrenheit in Centerville.
9. Determine the period of this function. What does this period represent?
10. What is the maximum high temperature and when does this occur?
© Glencoe/McGraw-Hill
840
NAME ______________________________________________ DATE
8-5
____________ PERIOD _____
Reading to Learn Mathematics
Graphing Trigonometric Functions
Pre-Activity Why can you predict the behavior of tides?
Read the introduction to Lesson 14-1 at the top of page 762 in your textbook.
Consider the tides of the Atlantic Ocean as a function of time.
Approximately what is the period of this function?
Reading the Lesson
1. Determine whether each statement is true or false.
a. The period of a function is the distance between the maximum and minimum points.
b. The amplitude of a function is the difference between its maximum and minimum
values.
c. The amplitude of the function y sin ␪ is 2.
d. The function y cot ␪ has no amplitude.
e. The period of the function y sec ␪ is .
f. The amplitude of the function y 2 cos ␪ is 4.
g. The function y sin 2␪ has a period of .
3
h. The period of the function y cot 3␪ is .
i. The amplitude of the function y 5 sin ␪ is 5.
1
4
j. The period of the function y csc ␪ is 4.
k. The graph of the function y sin ␪ has no asymptotes.
l. The graph of the function y tan ␪ has an asymptote at ␪ 180.
m. When ␪ 360, the values of cos ␪ and sec ␪ are equal.
n. When ␪ 270, cot ␪ is undefined.
o. When ␪ 180, csc ␪ is undefined.
Helping You Remember
2. What is an easy way to remember the periods of y a sin b␪ and y a cos b␪?
© Glencoe/McGraw-Hill
841
NAME ______________________________________________ DATE
8-5
____________ PERIOD _____
Enrichment
Blueprints
Interpreting blueprints requires the ability to select and use trigonometric
functions and geometric properties. The figure below represents a plan for an
improvement to a roof. The metal fitting shown makes a 30 angle with the
horizontal. The vertices of the geometric shapes are not labeled in these
plans. Relevant information must be selected and the appropriate function
used to find the unknown measures.
Example
Find the unknown
measures in the figure at the right.
Roofing Improvement
top view
The measures x and y are the legs of a
right triangle.
5"
––
16
metal fitting
The measure of the hypotenuse
15
5
20
is in. in. or in.
16
16
16
y
cos 30
20
x
sin 30
20
y 1.08 in.
x 0.63 in.
16
–15"
16–
x
side view
30
y
5"
––
16
0.09"
13"
––
16
16
Find the unknown measures of each of the following.
1. Chimney on roof
2. Air vent
1'
4 –2
3. Elbow joint
1'
3 –4
C
x
A
2'
D
B
1'
9 –2
40
1'
1 –2
y
1'
7 –4
r
A
1'
1 –4
40
© Glencoe/McGraw-Hill
4'
842
t
NAME ______________________________________________ DATE
8-6
____________ PERIOD _____
Study Guide and Intervention
Translations of Trigonometric Graphs
Horizontal Translations When a constant is subtracted from the angle measure in a
trigonometric function, a phase shift of the graph results.
The horizontal phase shift of the graphs of the functions y a sin b(␪ h), y a cos b(␪ h),
and y a tan b(␪ h) is h, where b 0.
If h 0, the shift is to the right.
If h 0, the shift is to the left.
Phase Shift
Example
State the amplitude, period, and
y
1.0
1
␲
phase shift for y ⫽ ᎏᎏ cos 3 ␪ ⫺ ᎏᎏ . Then graph
2
2
the function.
冢
冣
0.5
| |
O
1
2
1
2
2
2
2
Period: or | b|
|3|
3
Phase Shift: h 2
Amplitude: a or 0.5
6
3
2
2
3
5
6
1.0
2
The phase shift is to the right since 0.
Exercises
State the amplitude, period, and phase shift for each function. Then graph the
function.
冣
y
y
2
2
O
90
2
冢
2. y tan ␪ 1. y 2 sin (␪ 60)
90
180
270
O
360
2
3
冢
1
2
3. y 3 cos (␪ 45)
3
2
2
2
2
冣
4. y sin 3 ␪ y
y
1.0
2
O
0.5
90
180
270
360
O
0.5
450
2
1.0
© Glencoe/McGraw-Hill
843
6
3
2
2
3
5
6
NAME ______________________________________________ DATE
8-6
____________ PERIOD _____
Study Guide and Intervention (continued)
Translations of Trigonometric Graphs
Vertical Translations When a constant is added to a trigonometric function, the graph
is shifted vertically.
Vertical Shift
The vertical shift of the graphs of the functions y a sin b(␪ h) k, y a cos b(␪ h) k,
and y a tan b(␪ h) k is k.
If k 0, the shift is up.
If k 0, the shift is down.
The midline of a vertical shift is y k.
Graphing
Trigonometric
Functions
Step 1
Step 2
Step 3
Step 4
Determine the vertical shift, and graph the midline.
Determine the amplitude, if it exists. Use dashed lines to indicate the maximum and
minimum values of the function.
Determine the period of the function and graph the appropriate function.
Determine the phase shift and translate the graph accordingly.
Example
State the vertical shift, equation of the midline, amplitude, and
period for y ⫽ cos 2␪ ⫺ 3. Then graph the function.
y
Vertical Shift: k 3, so the vertical shift is 3 units down.
2
1
The equation of the midline is y 3.
Amplitude: | a | | 1 | or 1
2
b
O
1
2
2
2
3
2
2
Period: or | |
| |
Since the amplitude of the function is 1, draw dashed lines
parallel to the midline that are 1 unit above and below the midline.
Then draw the cosine curve, adjusted to have a period of .
Exercises
State the vertical shift, equation of the midline, amplitude, and period for each
function. Then graph the function.
1
2
1. y cos ␪ 2
2. y 3 sin ␪ 2
y
y
3
2
1
O
⫺1
⫺2
1
␲
2
␲
© Glencoe/McGraw-Hill
3␲
2
O
⫺1
⫺2
⫺3
⫺4
⫺5
⫺6
2␲
844
␲
2
␲
3␲
2
2␲
NAME ______________________________________________ DATE
8-6
____________ PERIOD _____
Skills Practice
Translations of Trigonometric Graphs
State the amplitude, period, and phase shift for each function. Then graph the
function.
1. y sin (␪ 90)
2. y cos (␪ 45)
y
y
2
4
1
1
2
90⬚ 180⬚ 270⬚ 360⬚
O
冣
y
2
O
2
冢
3. y tan ␪ 90⬚ 180⬚ 270⬚ 360⬚
O
⫺1
⫺1
⫺2
⫺2
⫺2
⫺4
␲
2
␲
3␲
2
2␲
State the vertical shift, equation of the midline, amplitude, and period for each
function. Then graph the function.
4. y csc ␪ 2
5. y cos ␪ 1
y
y
6. y sec ␪ 3
y
6
2
2
4
O
180⬚ 360⬚ 540⬚ 720⬚
1
⫺2
2
O
⫺4
180⬚ 360⬚ 540⬚ 720⬚
⫺1
O
90⬚ 180⬚ 270⬚ 360⬚
⫺2
⫺6
State the vertical shift, amplitude, period, and phase shift of each function. Then
graph the function.
7. y 2 cos [3(␪ 45)] 2
8. y 3 sin [2(␪ 90)] 2
y
6
4
4
4
2
2
2
O
90⬚ 180⬚ 270⬚ 360⬚
⫺2
© Glencoe/McGraw-Hill
O
冣冥 2
y
y
6
O
4
冤 43 冢
9. y 4 cot ␪ ⫺2
90⬚ 180⬚ 270⬚ 360⬚
⫺4
⫺2
845
␲
2
␲
3␲
2
2␲
NAME ______________________________________________ DATE
8-6
____________ PERIOD _____
Practice
Translations of Trigonometric Graphs
State the vertical shift, amplitude, period, and phase shift for each function. Then
graph the function.
冢
1
2
2
冣
1. y tan ␪ 2. y 2 cos (␪ 30) 3
y
y
y
4
6
2
4
2
O
2
3. y 3 csc (2␪ 60) 2.5
2
3
2
2
4
O
180
360
540
720
2
ECOLOGY For Exercises 4–6, use the following information.
The population of an insect species in a stand of trees follows the growth cycle of a
particular tree species. The insect population can be modeled by the function
y 40 30 sin 6t, where t is the number of years since the stand was first cut in
November, 1920.
4. How often does the insect population reach its maximum level?
5. When did the population last reach its maximum?
6. What condition in the stand do you think corresponds with a minimum insect population?
BLOOD PRESSURE For Exercises 7–9, use the following information.
Jason’s blood pressure is 110 over 70, meaning that the pressure oscillates between a maximum
of 110 and a minimum of 70. Jason’s heart rate is 45 beats per minute. The function that
represents Jason’s blood pressure P can be modeled using a sine function with no phase shift.
7. Find the amplitude, midline, and period in seconds of the function.
8. Write a function that represents Jason’s blood
pressure P after t seconds.
Jason’s Blood Pressure
P
120
9. Graph the function.
Pressure
100
80
60
40
20
0
© Glencoe/McGraw-Hill
846
1
2
3
4
5 6
Time
7
8
9 t
NAME ______________________________________________ DATE
8-6
____________ PERIOD _____
Reading to Learn Mathematics
Translations of Trigonometric Graphs
Pre-Activity How can translations of trigonometric graphs be used to show
animal populations?
Read the introduction to Lesson 14-2 at the top of page 769 in your textbook.
According to the model given in your textbook, what would be the estimated
rabbit population for January 1, 2005?
Reading the Lesson
1. Determine whether the graph of each function represents a shift of the parent function
to the left, to the right, upward, or downward. (Do not actually graph the functions.)
a. y sin (␪ 90)
3
冢
b. y sin ␪ 3
冣
c. y cos ␪ d. y tan ␪ 4
2. Determine whether the graph of each function has an amplitude change, period change,
phase shift, or vertical shift compared to the graph of the parent function. (More than
one of these may apply to each function. Do not actually graph the functions.)
5
6
冢
a. y 3 sin ␪ 冣
b. y cos (2␪ 70)
c. y 4 cos 3␪
1
2
d. y sec ␪ 3
4
冢
冣
e. y tan ␪ 1
冢 13
6
冣
f. y 2 sin ␪ 4
Helping You Remember
3. Many students have trouble remembering which of the functions y sin (␪ ␣) and
y sin (␪ ␣) represents a shift to the left and which represents a shift to the right.
Using ␣ 45, explain a good way to remember which is which.
© Glencoe/McGraw-Hill
847
NAME ______________________________________________ DATE
8-6
____________ PERIOD _____
Enrichment
Translating Graphs of Trigonometric Functions
Three graphs are shown at the right:
y 3 sin 2␪
y 3 sin 2(␪ 30)
y 4 3 sin 2␪
y
y = 3 sin 2u
O
90
y = 3 sin 2(u – 30 )
Replacing ␪ with (␪ 30) translates
the graph to the right. Replacing y
with y 4 translates the graph
4 units down.
Example
u
180
y + 4 = 3 sin 2u
Graph one cycle of y ⫽ 6 cos (5␪ ⫹ 80⬚) ⫹ 2.
Step 1 Transform the equation into
the form y k a cos b(␪ h).
y
6
y 2 6 cos 5(␪ 16)
Step 2
y = 6 cos 5u
O
Step 2 Sketch y 6 cos 5␪.
–6
Step 3 Translate y 6 cos 5␪ to
obtain the desired graph.
y
72
u
Step 3
y 2 2 = 6 cos 5( u + 16 )
6
y = 6 cos 5(u + 16 )
O
56
–6
Sketch these graphs on the same coordinate system.
1. y 3 sin 2(␪ 45)
2. y 1 3 sin 2␪
3. y 5 3 sin 2(␪ 90)
On another piece of paper, graph one cycle of each curve.
4. y 2 sin 4(␪ 50)
5. y 5 sin (3␪ 90)
6. y 6 cos (4␪ 360) 3
7. y 6 cos 4␪ 3
8. The graphs for problems 6 and 7 should be the same. Use the sum
formula for cosine of a sum to show that the equations are equivalent.
© Glencoe/McGraw-Hill
848
NAME ______________________________________________ DATE
8-7
____________ PERIOD _____
Study Guide and Intervention
Trigonometric Identities
Find Trigonometric Values A trigonometric identity is an equation involving
trigonometric functions that is true for all values for which every expression in the equation
is defined.
Basic
Trigonometric
Identities
cos ␪
sin ␪
Quotient Identities
tan ␪ sin ␪
cos ␪
cot ␪ Reciprocal Identities
csc ␪ 1
sin ␪
sec ␪ cot ␪ Pythagorean Identities
cos2 ␪ sin2 ␪ 1
tan2 ␪ 1 sec2 ␪
cot2 ␪ 1 csc2 ␪
Example
1
cos ␪
1
tan ␪
11
Find the value of cot ␪ if csc ␪ ⫽ ⫺ᎏᎏ; 180⬚ ⬍ ␪ ⬍ 270⬚.
5
2
2
Trigonometric identity
cot ␪ 1 csc ␪
冢 151 冣
cot2 ␪ 1 2
11
5
Substitute for csc ␪.
121
25
96
2
cot ␪ 25
4兹苶6
cot ␪ 5
11
5
cot2 ␪ 1 Square .
Subtract 1 from each side.
Take the square root of each side.
4兹苶6
5
Since ␪ is in the third quadrant, cot ␪ is positive, Thus cot ␪ .
Exercises
Find the value of each expression.
1. tan ␪, if cot ␪ 4; 180
␪
270
3
5
3. cos ␪, if sin ␪ ; 0 ␪
1
3
90
3
7
90
4
3
␪
180
6. tan ␪, if sin ␪ ; 0 ␪
7
8
␪
180
8. sin ␪, if cos ␪ ; 270 ␪
12
5
␪
180
10. sin ␪, if csc ␪ ; 270
7. sec ␪, if cos ␪ ; 90
9. cot ␪, if csc ␪ ; 90
90
2
4. sec ␪, if sin ␪ ; 0 ␪
90
5. cos ␪, if tan ␪ ; 90
© Glencoe/McGraw-Hill
兹3
苶
2. csc ␪, if cos ␪ ; 0 ␪
6
7
9
4
849
360
␪
360
NAME ______________________________________________ DATE
8-7
____________ PERIOD _____
Study Guide and Intervention (continued)
Trigonometric Identities
Simplify Expressions The simplified form of a trigonometric expression is written as a
numerical value or in terms of a single trigonometric function, if possible. Any of the
trigonometric identities on page 849 can be used to simplify expressions containing
trigonometric functions.
Example 1
Simplify (1 ⫺ cos2 ␪) sec ␪ cot ␪ ⫹ tan ␪ sec ␪ cos2 ␪.
1
cos ␪
cos ␪
sin ␪
sin ␪
cos ␪
1
cos ␪
(1 cos2 ␪) sec ␪ cot ␪ tan ␪ sec ␪ cos2 ␪ sin2 ␪ cos2 ␪
sin ␪ sin ␪
2 sin ␪
Example 2
sec ␪ cot ␪
1 ⫺ sin ␪
csc ␪
1 ⫹ sin ␪
Simplify ᎏᎏ ⫺ ᎏᎏ .
1
cos ␪
1
sin ␪
cos ␪ sin ␪
sec ␪ cot ␪
csc ␪
1 sin ␪
1 sin ␪
1 sin ␪
1 sin ␪
1
1
(1 sin ␪) (1 sin ␪)
sin ␪
sin ␪
(1 sin ␪)(1 sin ␪)
1
1
1 1
sin ␪
sin ␪
1 sin2 ␪
2
cos ␪
2
Exercises
Simplify each expression.
sin ␪ cot ␪
sec ␪ tan ␪
tan ␪ csc ␪
sec ␪
2. 2
2
sin2 ␪ cot ␪ tan ␪
cot ␪ sin ␪
4. 5. cot ␪ sin ␪ tan ␪ csc ␪
tan ␪ cos ␪
sin ␪
6. 7. 3 tan ␪ cot ␪ 4 sin ␪ csc ␪ 2 cos ␪ sec ␪
8. 1. cos ␪
sec ␪ tan ␪
3. © Glencoe/McGraw-Hill
csc2 ␪ cot2 ␪
tan ␪ cos ␪
850
1 cos2 ␪
tan ␪ sin ␪
NAME ______________________________________________ DATE
8-7
____________ PERIOD _____
Skills Practice
Trigonometric Identities
Find the value of each expression.
4
5
1. sin ␪, if cos ␪ and 90
␪
3. sec ␪, if tan ␪ 1 and 0 ␪
180
兹2
苶
7. cos ␪, if csc ␪ 2 and 180
␪
␪
1
2
4. cos ␪, if tan ␪ and 0 ␪
90
5. tan ␪, if sin ␪ and 180
2
2. cos ␪, if tan ␪ 1 and 180
270 6. cos ␪, if sec ␪ 2 and 270
90
␪
270
8. tan ␪, if cos ␪ and 180
5
9. cos ␪, if cot ␪ and 90
␪
180
10. csc ␪, if cos ␪ and 0
11. cot ␪, if csc ␪ 2 and 180
␪
270
12. tan ␪, if sin ␪ and 180
8
17
5
13
Simplify each expression.
13. sin ␪ sec ␪
14. csc ␪ sin ␪
15. cot ␪ sec ␪
16. 17. tan ␪ cot ␪
18. csc ␪ tan ␪ tan ␪ sin ␪
cos ␪
sec ␪
1 sin2 ␪
sin ␪ 1
20. csc ␪ cot ␪
sin2 ␪ cos2 ␪
1 cos
22. 1 19. 21. 2 ␪
© Glencoe/McGraw-Hill
360
苶
2 兹5
␪
3
2
270
tan2 ␪
1 sec ␪
851
␪
␪
270
90
␪
270
NAME ______________________________________________ DATE
8-7
____________ PERIOD _____
Practice
Trigonometric Identities
Find the value of each expression.
5
13
1. sin ␪, if cos ␪ and 0 ␪
90
15
17
2. sec ␪, if sin ␪ and 180
␪
3
10
␪
360
4. sin ␪, if cot ␪ and 0 ␪
90
3
2
␪
270
6. sec ␪, if csc ␪ 8 and 270
␪
360
␪
360
3. cot ␪, if cos ␪ and 270
5. cot ␪, if csc ␪ and 180
7. sec ␪, if tan ␪ 4 and 180
2
5
␪
9. cot ␪, if tan ␪ and 0 ␪
270
90
1
2
270
1
2
8. sin ␪, if tan ␪ and 270
1
3
10. cot ␪, if cos ␪ and 270
␪
360
Simplify each expression.
sin2 ␪
tan ␪
13. sin2 ␪ cot2 ␪
csc2 ␪ cot2 ␪
1 cos ␪
16. cos ␪
1 sin ␪
19. sec2 ␪ cos2 ␪ tan2 ␪
11. csc ␪ tan ␪
12. 2
14. cot2 ␪ 1
15. 2
17. sin ␪ cos ␪ cot ␪
18. cos ␪
1 sin ␪
csc ␪ sin ␪
cos ␪
20. AERIAL PHOTOGRAPHY The illustration shows a plane taking
an aerial photograph of point A. Because the point is directly below
the plane, there is no distortion in the image. For any point B not
directly below the plane, however, the increase in distance creates
distortion in the photograph. This is because as the distance from
the camera to the point being photographed increases, the
exposure of the film reduces by (sin ␪)(csc ␪ sin ␪). Express
(sin ␪)(csc ␪ sin ␪) in terms of cos ␪ only.
A
B
21. TSUNAMIS The equation y a sin ␪t represents the height of the waves passing a
buoy at a time t in seconds. Express a in terms of csc ␪t.
© Glencoe/McGraw-Hill
852
NAME ______________________________________________ DATE
8-7
____________ PERIOD _____
Reading to Learn Mathematics
Trigonometric Identities
Pre-Activity How can trigonometry be used to model the path of a baseball?
Read the introduction to Lesson 14-3 at the top of page 777 in your textbook.
Suppose that a baseball is hit from home plate with an initial velocity of
58 feet per second at an angle of 36 with the horizontal from an initial
height of 5 feet. Show the equation that you would use to find the height of
the ball 10 seconds after the ball is hit. (Show the formula with the
appropriate numbers substituted, but do not do any calculations.)
Reading the Lesson
1. Match each expression from the list on the left with an expression from the list on the
right that is equal to it for all values for which each expression is defined. (Some of the
expressions from the list on the right may be used more than once or not at all.)
1
sin ␪
a. sec2 ␪ tan2 ␪
i. b. cot2 ␪ 1
ii. tan ␪
sin ␪
cos ␪
c. iii. 1
d. sin2 ␪ cos2 ␪
iv. sec ␪
e. csc ␪
v. csc2 ␪
1
cos ␪
vi. cot ␪
f. cos ␪
sin ␪
g. 2. Write an identity that you could use to find each of the indicated trigonometric values
and tell whether that value is positive or negative. (Do not actually find the values.)
4
5
␪
a. tan ␪, if sin ␪ and 180
b. sec ␪, if tan ␪ 3 and 90
␪
270
180
Helping You Remember
3. A good way to remember something new is to relate it to something you already know.
How can you use the unit circle definitions of the sine and cosine that you learned in
Chapter 13 to help you remember the Pythagorean identity cos2 ␪ sin2 ␪ 1?
© Glencoe/McGraw-Hill
853
NAME ______________________________________________ DATE
8-7
____________ PERIOD _____
Enrichment
Planetary Orbits
The orbit of a planet around the sun is an ellipse with
the sun at one focus. Let the pole of a polar coordinate
system be that focus and the polar axis be toward the
other focus. The polar equation of an ellipse is
r
b2
c
2ep
1 e cos ␪
r . Since 2p and b2 a2 c2,
冢
冣
a2 c2
c
a2
c
冢 ac 冣冢
冢 ac 冣 冣 a冢1e冣(1 e2).
c2
a
c
a
2p 1 2 . Because e ,
2p a 1 2
Therefore 2ep a(1 e2). Substituting into the polar equation of an
ellipse yields an equation that is useful for finding distances from the
planet to the sun.
a(1 e2)
1 e cos ␪
r Note that e is the eccentricity of the orbit and a is the length of the
semi-major axis of the ellipse. Also, a is the mean distance of the planet
from the sun.
Example
The mean distance of Venus from the sun is
67.24 ⫻ 106 miles and the eccentricity of its orbit is .006788. Find the
minimum and maximum distances of Venus from the sun.
The minimum distance occurs when ␪ .
67.24 106(1 0.0067882)
1 0.006788 cos r 66.78
106 miles
The maximum distance occurs when ␪ 0.
67.24 106(1 0.0067882)
1 0.006788 cos 0
r 67.70
106 miles
Complete each of the following.
1. The mean distance of Mars from the sun is 141.64 106 miles and the
eccentricity of its orbit is 0.093382. Find the minimum and maximum
distances of Mars from the sun.
2. The minimum distance of Earth from the sun is 91.445 106 miles and
the eccentricity of its orbit is 0.016734. Find the mean and maximum
distances of Earth from the sun.
© Glencoe/McGraw-Hill
854
Polar Axis
NAME ______________________________________________ DATE
8-8
____________ PERIOD _____
Study Guide and Intervention
Verifying Trigonometric Identities
Transform One Side of an Equation Use the basic trigonometric identities along
with the definitions of the trigonometric functions to verify trigonometric identities. Often it
is easier to begin with the more complicated side of the equation and transform that
expression into the form of the simpler side.
Example
Verify that each of the following is an identity.
tan ␪
csc ␪
sin ␪
cot ␪
a. ᎏ ⫺ sec ␪ ⫽ ⫺cos ␪
b. ᎏ ⫹ cos ␪ ⫽ sec ␪
Transform the left side.
Transform the left side.
sin ␪
sec ␪ ⱨ cos ␪
cot ␪
tan ␪
cos ␪ ⱨ sec ␪
csc ␪
sin ␪
1
cos ␪ cos ␪ ⱨ cos ␪
sin ␪
cos ␪
cos ␪ ⱨ sec ␪
1
sin ␪
sin ␪
sin2 ␪
1
ⱨ cos ␪
cos ␪
cos ␪
sin2 ␪
cos ␪ ⱨ sec ␪
cos ␪
sin2 1
ⱨ cos ␪
cos ␪
sin2 ␪ cos2 ␪
ⱨ sec ␪
cos ␪
cos2 ␪
ⱨ cos ␪
cos ␪
1
ⱨ sec ␪
cos ␪
cos ␪ cos ␪
sec ␪ sec ␪
Exercises
Verify that each of the following is an identity.
1. 1 csc2 ␪ cos2 ␪ csc2 ␪
© Glencoe/McGraw-Hill
sin ␪
1 cos ␪
cot ␪
1 cos ␪
1 cos3 ␪
sin ␪
2. 3
855
NAME ______________________________________________ DATE
8-8
____________ PERIOD _____
Study Guide and Intervention (continued)
Verifying Trigonometric Identities
Transform Both Sides of an Equation The following techniques can be helpful in
verifying trigonometric identities.
• Substitute one or more basic identities to simplify an expression.
• Factor or multiply to simplify an expression.
• Multiply both numerator and denominator by the same trigonometric expression.
• Write each side of the identity in terms of sine and cosine only. Then simplify each side.
Example
tan2 ␪
1
Verify that ᎏᎏᎏ
⫽ sec2 ␪ ⫺ tan2 ␪ is an identity.
sin ␪ tan ␪ sec ␪ 1
tan2 ␪ 1
ⱨ sec2 ␪ tan2 ␪
sin ␪ tan ␪ sec ␪ 1
sec2 ␪
1
sin2 ␪
ⱨ 2 ␪ 2 ␪
sin ␪
1
cos
cos
sin ␪ 1
cos ␪
cos ␪
1
cos2 ␪
1 sin2 ␪
ⱨ 2
sin ␪
cos2 ␪
1
2
cos ␪
1
cos2 ␪
cos2 ␪
ⱨ
2
2
sin ␪ cos ␪
cos2 ␪
2
cos ␪
1
ⱨ1
sin2 ␪ cos2 ␪
11
Exercises
Verify that each of the following is an identity.
tan2 ␪
1 cos ␪
1. csc ␪ sec ␪ cot ␪ tan ␪
cos ␪ cot ␪
sin ␪
csc ␪
sin ␪ sec ␪
csc2 ␪ cot2 ␪
sec ␪
3. 2
© Glencoe/McGraw-Hill
sec ␪
cos ␪
2. 2
4. cot2 ␪(1 cos2 ␪)
2
856
NAME ______________________________________________ DATE
8-8
____________ PERIOD _____
Skills Practice
Verifying Trigonometric Identities
Verify that each of the following is an identity.
1. tan ␪ cos ␪ sin ␪
2. cot ␪ tan ␪ 1
3. csc ␪ cos ␪ cot ␪
4. cos ␪
5. (tan ␪)(1 sin2 ␪) sin ␪ cos ␪
6. cot ␪
1 sin2 ␪
cos ␪
csc ␪
sec ␪
2
sin2 ␪
1 sin ␪
cos2 ␪
1 sin ␪
7. tan2 ␪
2
© Glencoe/McGraw-Hill
8. 1 sin ␪
857
NAME ______________________________________________ DATE
8-8
____________ PERIOD _____
Practice
Verifying Trigonometric Identities
Verify that each of the following is an identity.
cos2 ␪
1 sin ␪
sin2 ␪ cos2 ␪
cos ␪
1. sec2 ␪
2
2. 1
2
3. (1 sin ␪)(1 sin ␪) cos2 ␪
4. tan4 ␪ 2 tan2 ␪ 1 sec4 ␪
5. cos2 ␪ cot2 ␪ cot2 ␪ cos2 ␪
6. (sin2 ␪)(csc2 ␪ sec2 ␪) sec2 ␪
7. PROJECTILES The square of the initial velocity of an object launched from the ground is
2gh
sin ␪
v2 2 , where ␪ is the angle between the ground and the initial path, h is the
maximum height reached, and g is the acceleration due to gravity. Verify the identity
2gh sec2 ␪
2gh
.
sin2 ␪
sec2 ␪ 1
8. LIGHT The intensity of a light source measured in candles is given by I ER2 sec ␪,
where E is the illuminance in foot candles on a surface, R is the distance in feet from the
light source, and ␪ is the angle between the light beam and a line perpendicular to the
surface. Verify the identity ER2(1 tan2 ␪) cos ␪ ER2 sec ␪.
© Glencoe/McGraw-Hill
858
NAME ______________________________________________ DATE
8-8
____________ PERIOD _____
Reading to Learn Mathematics
Verifying Trigonometric Identities
Pre-Activity How can you verify trigonometric identities?
Read the introduction to Lesson 14-4 at the top of page 782 in your textbook.
For ␪ , 0, or , sin ␪ sin 2␪. Does this mean that sin ␪ sin 2␪ is an
identity? Explain your reasoning.
Reading the Lesson
1. Determine whether each equation is an identity or not an identity.
1
sin ␪
1
tan ␪
a. 1
2
2
cos ␪
sin ␪ tan ␪
b. sin ␪
cos ␪
cos ␪
sin ␪
c. cos ␪ sin ␪
d. cos2 ␪ (tan2 ␪ 1) 1
sin2 ␪
cos ␪
sin ␪ csc ␪ sec2 ␪
e. 2
1
1 sin ␪
1
1 sin ␪
f. 2 cos2 ␪
1
csc ␪
g. tan2 ␪ cos2 ␪ 2
sin ␪
sec ␪
1
tan ␪
1
cot ␪
h. 2. Which of the following is not permitted when verifying an identity?
A. simplifying one side of the identity to match the other side
B. cross multiplying if the identity is a proportion
C. simplifying each side of the identity separately to get the same expression on both sides
Helping You Remember
3. Many students have trouble knowing where to start in verifying a trigonometric identity.
What is a simple rule that you can remember that you can always use if you don’t see a
quicker approach?
© Glencoe/McGraw-Hill
859
NAME ______________________________________________ DATE
8-8
____________ PERIOD _____
Enrichment
Heron’s Formula
Heron’s formula can be used to find the area of a triangle if you know the
lengths of the three sides. Consider any triangle ABC. Let K represent the
area of 䉭ABC. Then
1
2
K bc sin A
B
b2c2 sin2 A
4
K 2 c
Square both sides.
b2c2(1 cos2 A)
4
A
a
C
b
b2c2(1 cos A)(1 cos A)
4
冢
b2 c2 a2
2bc
bca
2
bca
2
b2c2
4
冣冢
b2 c2 a2
2bc
1 1 abc
2
冣
Use the law of cosines.
abc
2
Simplify.
abc
2
bca
2
acb
2
abc
2
Let s . Then s a , s b , s c .
K 2 s(s a)(s b)(s c)
Substitute.
K 兹苶
s(s 苶
a)(s 苶
b)(s 苶
c)
Heron’s Formula
The area of 䉭ABC is
兹s(s a)(s b)(s c), where s abc
.
2
Use Heron’s formula to find the area of 䉭ABC.
1. a 3, b 4.4, c 7
2. a 8.2, b 10.3, c 9.5
3. a 31.3, b 92.0, c 67.9
4. a 0.54, b 1.32, c 0.78
5. a 321, b 178, c 298
6. a 0.05, b 0.08, c 0.04
7. a 21.5, b 33.0, c 41.7
8. a 2.08, b 9.13, c 8.99
© Glencoe/McGraw-Hill
860