# Right Triangles and Trigonometry (Geo. Sketchpad)

advertisement ```Table of Contents
Part I: Unit Plan…………………………………………………..3-8
1. Unit Title
2. Unit Summary
3. Key Words
4. Background Knowledge
5. NCTM Standard(s) Addressed
a. State Standards, Benchmarks and
Grade Level Indicators
6. Learning Objectives
7. Materials
8. Suggested Procedures
a. Attention Getter
b. Suggested Grouping
9. Assessment(s)
Part II: Inquiry-based Activities………………………….9-57
Lesson One:
Lesson Two:
The Geometric Mean w/homework
Visual Proof of Pythagorean Theorem
w/homework
Lesson Three: Investigations into the Converse of
the Pythagorean Theorem w/homework
Lesson Four: Special right triangles w/homework
Lesson Five: Investigations of trig ratios
w/homework
Lesson Six:
Investigations of Inverse Trig
Functions w/homework
Part III: Solutions ………………………………………….58-73
Unit Title
Investigations, explorations, and applications of right
triangles and trigonometry
Lesson Summary
The inquiry-based activities in this lesson include formulating and
testing ideas involving right triangles. Students will employ a variety of
problem-solving techniques including using trigonometry, indirect
measurements, constructions and the concept of geometric mean.
Students will begin by exploring similar right triangles and the
geometric mean. Students will then review the Pythagorean Theorem,
formulate and test its converse, investigate Pythagorean triples, 4545-90 and 30-60-90 special right triangles. Finally students will
generalize formulas to solve right triangles and their real-world
applications by correct selection and use of the tangent, sine and
cosine ratios.
Key Words
Right triangles, acute triangles, obtuse triangles, trigonometry,
geometric mean, special right triangles, Converse of the Pythagorean
Theorem, leg, hypotenuse, adjacent segment, altitude, and
trigonometric ratios.
Background Knowledge
Prior to this lesson students should have substantial knowledge and
skills in the following areas:
 Properties of right triangles
 Applying the Pythagorean Theorem
 Finding converses given conditional statements
 Differentiating between &lt; and &gt;inequalities
 Ratios and Proportions
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Determining similarity of triangles
Finding squares and square roots
Definition of geometric mean
Construction of a triangle given three sides using a compass
and straightedge
 Geometer's Sketchpad or Cabri
 Recognizing complementary angles
 Using the trigonometric function keys on the calculator
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Ohio Standards Addressed
Ohio Content Standards Addressed: Number, Number Sense and
Operations, Measurement, Geometry and Spatial Sense, Patterns,
Functions and Algebra
Benchmarks- Number Sense
8-10 D: Connect physical, verbal and symbolic representations of
integers, rational numbers and irrational numbers.
8-10 E: Compare, order and determine equivalent forms of real
numbers.
8-10 H: Find the square root of perfect squares, and approximate the
square root of non-perfect squares
11-12 E: Represent and compute with complex numbers.
Benchmarks- Measurement
8-10 E: Estimate and compute various attributes, including length, angle
measure, area, surface area and volume to a specified level of
precision.
8-10 G: Use proportional reasoning and apply indirect measuring
techniques, including right triangle trigonometry and properties of
similar triangles to solve problems involving measurements and rates.
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Benchmarks- Geometry and Spatial Sense
8-10 B: Describe and apply the properties of similar and
congruent figures; and justify conjectures involving similarity
and congruence.
8-10 E: Draw and construct representations of two-and threedimensional geometric objects using a variety of tools, such as
straight edge, compass and technology.
8-10 H: Establish the validity of conjectures about geometric
objects their properties and relationships by counterexamples,
inductive and deductive reasoning and critiquing arguments made
by others
8-10 I: Use right triangle trigonometric relationships to
determine length and angle measures.
11-12A: Use trigonometric relationships to verify and determine
solutions in problem situations.
Benchmarks-Patterns, Functions and Algebra
8-10 B: Identify and classify functions as linear or non-linear and
contrast their properties using tables, graphs or equations.
8-10 D. Use algebraic representations such as tables, graphs,
expressions, functions and inequalities to model and solve problem
situations.
Grade level Indicators – Number Sense Standard
E 9-2: Compare, order and determine equivalent forms for rational and
irrational numbers.
D 10-1: Connect physical, verbal and symbolic representations of
irrational numbers; e.g., construct the square root of 2 as a hypotenuse
or on a number line.
E 11-7: Compute sums, differences, products and quotients of complex
numbers.
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Grade level Indicators – Measurement Standard
E 10-6: Estimate lengths of missing segments and measurements of
missing angles using trigonometric charts and tables and interpolation
to a specified number of significant digits.
G 9-4: Use scale drawings, properties of similar polygons, and right
triangle trigonometry to solve problems that include unknown distances
and angle measurements.
Grade level Indicators – Geometry and Spatial Sense Standard
H 10-3: Prove the Pythagorean Theorem.
I 9-1: Define the basic trigonometric ratios in right triangles: sine,
cosine, and tangent.
I 9-2: Solve right triangle problems by correct selection and use of the
tangent, sine and cosine ratios.
Grade level Indicators – Patterns, Functions and Algebra Standard
B 8-9: Solve and use linear inequalities to describe parameters of
geometric figures.
D 8-8: Determine the lengths of two sides of special right triangles
when the length of the third side is known.
Learning Objectives
Upon completion of this lesson, students will demonstrate knowledge
and be able to:
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 State and apply the relationships that exist when the altitude is
drawn to the hypotenuse of a right triangle
 State and apply the Pythagorean Theorem, its converse, and
related theorems about obtuse and acute triangles
 Recognize Pythagorean triples and their multiples in right triangle
problems
 Determine the lengths of two sides of a 45-45-90 or a 30-60-90
triangle when the length of a third side is given
 Define the tangent, sine, and cosine ratios for both the acute
angles in a right triangle
 State and apply the relationship that exists between the trig
functions of an acute angle of a right triangle and those of its
complement
 Recognize the secant, cosecant and cotangent functions as the
inverse functions of sine, cosine and tangent
Materials
All handouts are included in this packet. Students will need pencil and
eraser, a straightedge and a compass, a scientific or graphing
calculator and access to dynamic geometric software. For the visual
proof in Activity Two, they will also need scissors and tape/glue or
sticky-back paper.
Suggested Procedures
a. See individual lesson plans for “attention getters”.
b. Unless otherwise noted in the activity, small groups are
recommended (3-4 students per group).
Assessments
Both formal and informal assessments will be utilized:
 In-class handouts
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Contributions to group effort
Peer evaluation
Participation in class discussion
Daily homework
Quizzes
Test
Projects
Lesson One
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Exploring the relationships created by the altitude of a
right triangle.
Lesson summary: The students will explore the similar right triangles and geometric mean
relationships created when the altitude from the right angle to the hypotenuse of a right
triangle is drawn. First the students determine what angles of the three triangles (the original
right triangle, and the two on each side of the altitude) are congruent. Then they use the
congruent angles to determine similarity relationships between the three triangles. Next, they
use the similar triangles to write proportion statements relating the sides of the triangle.
Finally, the students will determine which proportion statements are geometric means.
Key words: Altitude of right triangle, and geometric mean.
Background knowledge: Definition of a geometric mean; Properties of similar triangles;
Proving triangles similar; Parts of a right triangle; Definition of an altitude.
Standards: See overall project description.
Learning Objectives: The student will …
 Identify the similar triangles created when an altitude is drawn from the right angle of
a right triangle to the hypotenuse.
 Realize the altitude of the right triangle is the geometric mean of the two segments of
the hypotenuse created by the altitude.
 Realize that each leg of a right triangle is the geometric mean of the hypotenuse and
the segment of the hypotenuse created by the altitude that is adjacent to the leg.
Materials: Handout, tape, chalk or other materials to create or draw a large right triangle and the
altitude of that right triangle.
Suggested procedures:
“Attention getter”: Construct or draw a large right triangle (3 feet by 4 feet by 5 feet, for
example) with altitude on the board, a wall, the floor or ceiling. First discuss why
there appears to be only 1 altitude for this triangle (the other 2 altitudes are the legs of
the triangle). Then have the students identify the legs, altitude, and hypotenuse of the
right triangle. Then discuss the two segments of the hypotenuse that the altitude
creates, and determine which leg is adjacent to each of these segments.
Groups: groups of 3-4 students are recommended. Random selection, heterogeneous skill
level groups or groups that are previously established can be used.
Assessment: The worksheet for the activity should be collected. Homework problems are
provided. Test/quiz questions are also recommended.
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Name: _____________________
Lesson One: Exploring the relationships created by the altitude of a
right triangle
Goal: Students will explore similar right triangles and geometric mean relationships created
when the altitude from the right angle to the hypotenuse of a right triangle is drawn.
A
Given a right triangle, when the altitude is
drawn from the vertex of the right angle to
the hypotenuse, two new triangles are formed.
Below, these three triangles are drawn side
by side. The sides a, b, and c from the original
triangle are labeled, along with the two
segments of the hypotenuse (d and e) that are
created by the altitude (f).
D
C
A
B
A
C
c
a
f
b
d
b
D
C
a
B
B
e
D
f
C
1. List the corresponding congruent angles between ABC and CBD .
2. Now consider ABC and ACD . List their congruent corresponding angles.
3. Finally, write the corresponding angle congruence statements for CBD and ACD .
4. If two or more angles of a triangle are congruent to two or more angles in a second triangle,
then the triangles are similar. List all sets of similar triangles above.
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A
c
a
f
b
d
b
D
C
a
B
e
D
B
f
C
5. Since all three triangles are similar to each other, we can write proportions relating sets of
corresponding sides. Write a proportion that contains side f twice (the altitude from the right
angle of ABC ).
6. By Definition, x is the geometric mean of m and n if
x n m x
 or  .
m x
x n
Are any of the segments in the above proportion a geometric mean? Write a definition for the
geometric mean you discovered in question 5, using the names of the segments related to the
original triangle (leg, hypotenuse, altitude, segments of the hypotenuse).
7. Write a proportion that contains side b twice (one of the legs of ABC ).
8. Write a definition for the geometric mean you discovered in question 5, using the names of
the segments related to the original triangle (leg, hypotenuse, altitude, segment of the
hypotenuse adjacent to the leg, segment of the hypotenuse not adjacent to the leg).
9. Write a proportion that contains side a twice (the other leg of ABC )
10. Can you use the same definition for the geometric mean relationship from 8 to describe the
geometric mean relationship related to side a from question 9?
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Name:__________________
Lesson One : Homework
Solve for all unknowns. Show work for any credit.
1.
4
Y
X
3
Z
2.
4
6
L
M
N
Extension: Find the value of X.
4
6
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X
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Lesson Plan
Lesson Two – Revisiting the Pythagorean Theorem – A Visual Proof
Lesson Summary This lesson leads students through a visual proof of the
Pythagorean Theorem using a provided pattern. In addition, students are asked to
derive both a verbal and algebraic statement of this extensively used theorem.
Background knowledge Prior to this lesson students should have substantial
knowledge and skills in the following areas:
 Properties of right triangles
 Applying the Pythagorean Theorem
 Finding converses given conditional statements
Ohio Standards/Benchmarks/Grade Level Indicators Addressed
Benchmarks- Geometry and Spatial Sense - 8-10 H: Establish the validity of
conjectures about geometric objects their properties and relationships by
counterexamples, inductive and deductive reasoning and critiquing arguments made
by others. Grade Level Indicator H 10-3: Prove the Pythagorean Theorem.
Learning Objectives Students will be able to verbally state, algebraically
represent and visually prove the Pythagorean Theorem.
Materials Needed
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paper, pencil, eraser, straightedge, scissors, glue/tape
Individual copies of the handout
Individual copy of &quot;net&quot; copied onto sticky back paper
Individual copy of &quot;net&quot; with 2-column proof of the Pythagorean Theorem on
the reverse side (This one will be saved and put into the student's notes)
2 overhead transparencies (incomplete and completed 2-column proof)
Overhead projector or S-video or Smartboard and screen
Suggested Procedures
The following require preparation in advance: 1) Individual student copies of the
packet and the &quot;net,&quot; and two overhead transparencies. 2) Predetermination of
preferred grouping (2 to 4 suggested). The remaining procedures are suggested:
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 &quot;Attention Grabber&quot;-Place the transparency of the incomplete 2-column
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proof of the Pythagorean Theorem (provided) on the overhead projector.
Have it projected onto the screen as students enter the room. Meet them at
the door and excitedly tell them that TODAY is the day they are going to
PROVE the Pythagorean Theorem. (Listen to them groan!)
Transition from Activity One (Geometric Mean) by asking students if they
recognize the diagram used for this formal proof.
Review yesterday's activity through several quick questions.
Refocus the students' attention to the Proof. Elicit ideas on how to start
and finalize the proof.
Give in reluctantly and place the completed proof on the overhead. You might
have one or two students come to the front and read the statements and
reasons of the proof to the class.
Interrupt the students long enough to distribute the packet so that the
students can follow along.
Direct the students to underline all the algebraic equations in the proof.
When the proof has been presented in its entirety, tell the students, &quot;Now
it's your turn!&quot; Divide the students into their predetermined groups' point
out where the activity starts and then walk around the class assisting as
needed.
At the end of class bring the students back to their &quot;whole class&quot; seating,
take a quick oral survey of how they did, and distribute and explain the
homework and the plan for the next day.
Assessments
Informal: observation during group work, questioning, participation in review
Formal: Check answers to the activity, homework, quiz after Activity 3, and
Test at end of unit.
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Lesson Two:
&quot;Now You See It …Don't Say You Don't.&quot;
Name: ___________________________ Per ____ Date ________
Steps for proof:
- Cut out the smaller square (number 5) and parts 1-4 of the middle square in the attached net.
- Arrange these pieces to exactly cover the larger square ABCD.
- When you are absolutely sure that you have accomplished this:
a) lay the pieces out on the second (uncut) sheet,
b) trace the outline of these pieces onto the largest square
c) peel back the sticky tab and permanently place the pieces on the second (uncut) sheet.
*** When you have successfully completed the above steps, you have demonstrated the area of
the square on the hypotenuse is equal to the sum of the areas of the squares on the two legs of
your triangle.
1) Compare your results with the results of other groups near you.
2) If the lengths of the two legs of a right triangle are named a and b, then the areas of the
squares on the legs would be ________ and ___________. (Refer to the diagram on the uncut
sheet).
3) If the length of the hypotenuse is c, then the area of the square of the hypotenuse is ________.
4) Using the visual diagram, combine the results from steps 4 and 5 into an equation and write it
on the line below.
_____________________________________
5) What have you just discovered?
6) Rewrite the concept you have just proven as a conditional statement in &quot;if…then&quot; form.
7) How does this visual help to make sense out of the famous theorem?
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Do not discard this activity and especially your square. We will use this later!!!
Name: ____________________
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Homework for Pythagorean Theorem
1) The classical ladder problem:
There is a building with a 12 ft high window. You want to use a ladder to go up to the
window, and you decide to keep the ladder 5 ft away from the building to have a good
slant. How long should the ladder be?
2) Baseball diamond:
On a baseball diamond the bases are 90 ft apart. What is the distance from home plate
to second base in a straight line?
3) An algebraic problem:
Find the length of both of the missing sides on the following triangle:
4) An iterative problem:
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Look at the following figure. Start by finding the value for X 1, then for X2, then X3,
and so on until you get the value for X6. Write the lengths as square roots, as that
makes it simpler.
What is the value of X6?
Extension:
5) Equilateral Triangle:
An equilateral triangle has vertices (0,0) and (6,0) in a coordinate plane. What are
the coordinates of the third vertex? You may want to sketch it out.
Note: the sides of an equilateral triangle are equal in length.
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LESSON PLAN THREE
Lesson Three – Exploring the Converse of the Pythagorean Theorem
and its Corollaries
Lesson Summary: This lesson is combined with lesson four to allow students to engage in a
two-tiered activity in order to derive and apply the Converse of the Pythagorean Theorem.
Students will construct triangles, and measure, calculate, reason, conjecture and finally justify
the conclusion that triangles whose sides satisfy the equation a2 + b2 = c2 are in fact right
triangles. They will also conjecture about the types of triangles formed when a2 + b2 &gt; c2 and
will work with Pythagorean Triples.
Background Knowledge: Students will need to be familiar with: Properties of right triangles,
applying the Pythagorean Theorem, finding converses given conditional statements, finding
squares and square roots, the geometric mean, construction of a triangle given three sides using a
compass and straightedge, Geometer's Sketchpad or Cabri.
Ohio Standards Addressed: 8-10 G: Use proportional reasoning and apply indirect measuring
techniques, including right triangle trigonometry and properties of similar triangles to solve
problems involving measurements and rates.
Learning Objectives: Upon completion of this lesson, students will demonstrate knowledge and
be able to:
 State and apply the Pythagorean Theorem, its converse, and related theorems about
obtuse and acute triangles
 Recognize Pythagorean triples and their multiples in right triangle problems
Materials Needed: Pencil, paper, eraser, straightedge, ruler, compass, and handout.
Technology required: Access to student computers loaded with a dynamic Geometry software
such as Geometer's Sketchpad or Cabri. Optional: S-video, Internet
Suggested Procedures:
 Requires advanced preparation: Installation of software, scheduled use of student
computers, individual student copies of packet for Activity Three, determination of
grouping for students (2 to 4 per group, 1 to 2 per computer).
 &quot;Attention Grabber&quot; Write on the board or overhead for the students to complete in
their journals: &quot;If [Mrs. Millin] had a dollar for ever Geometry student who hated proofs,
then……………&quot;
 Answer questions on the previous night's homework. Have students put 2 or 3 of the
problems on the board. Have students check their work for accuracy. Collect.
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 Generate a discussion of conditional statements and their converses with voluntary

student conditionals. Ask: &quot;Is this conditional statement true? Is its converse true? &quot;Is
there a general rule about conditionals and their converses?&quot;
Ask about the conditional statement the students wrote on Question # 6 of yesterday's
class activity. &quot;Who is pretty sure he wrote the converse of the Pythagorean Theorem
correctly? Be careful: It's tricky!&quot;
 Have a student write the converse on the board being sure to use the correct wording.
Instruct students to get out their compasses, straightedges and rulers while you distribute
the packet.
 Transition to today's activity through a summary of the concepts explored yesterday.
 Explain to the students that this activity will be combined with a second activity that will
take two days and the second day they will be using computers. Dismiss them to their
predetermined groups.
 Walk around the classroom assisting students as needed. Observe students level of
involvement and cooperation within their groups for assessment purposes.
 Stop three or four minutes before the bell. Give students their homework
Assessments
Informal: Observation during group work, questioning, participation in review.
Formal: Check answers to the activity, homework, quiz after Activity 3, and
Test at end of unit.
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Name: __________________
Lesson Three: If a triangle's not right, then it's…
Lesson Goals: Through measurement, constructions omit and calculations, and using
their knowledge of the Pythagorean Theorem, the students will identify right,
acute, and obtuse triangles based on the length of their sides,
RECALL the converse of the Pythagorean Theorem from today's discussion and write it out in
the space below:
How to construct a triangle:
Example for you to trace over and practice on: Example: Construct a triangle with side lengths of
(1, 3 , 2) in inches ( 3  1.73)
Steps:
1, Draw a segment longer
than the longest side and then
mark off two endpoints for
that length.
2. Set your compass at the length of one of the remaining sides and draw a semicircle
using one of the endpoints you marked on the segment as the center.
3. Reset your compass to the length of the remaining side and using the other endpoint as
the new center, draw a semicircle.
4. The point of intersection of the two arcs is the third vertex of your triangle. (The
endpoints of your longest segment are the other two.
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1. DRAW the following triangles with sides as given. (Use cms for your unit of measurement)
 CAT (3,4,5)
 SIT(9,12,15)
 DOG (6,8,10)
 RUN (5,12,13)
2. For each of the triangles in Exercise 1 compare the sum of the squares of the lengths of the
two shorter sides with the square of the length of the longest side (using &lt;, &gt; or =).
Example:  HOW (15, 20, 25)  152 + 202 ? 252  225 + 400 = 625
 CAT (3, 4, 5)
 DOG (6, 8,10)
(__)2 + (__)2 _?_ (__)2
(__)2 + (__)2 _?_ (__)2
____ + ____ ___ ____
____ + ____ ___ ____
 SIT (9, 12, 15)
 RUN (5, 12, 13)
(__)2 + (__)2 _?_ (__)2
(__)2 + (__)2 _?_ (__)2
____ + ____ ___ ____
____ + ____ ___ ____
3. Have a teammate check the accuracy of your calculations. _______What do your
Initials
calculations in exercise 2 suggest about all of the triangles you constructed?
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4. Verify your answer by using your protractor to measure (to the nearest degree) the angle
opposite the longest side length in each of the triangles in Exercise 1. Write your angle
measurements inside the angles.
Have a teammate check the accuracy of your measurements. ______
Initials
5. Do you think your findings in Exercises 2 and 4 are true …for all triangles? (Yes, NO) … or
for only certain triangles? (Yes, NO). Explain your reasoning.
6. Using only your straightedge and a compass, construct  s with the given side lengths (in
inches or centimeters).
 CBS (2,3,3)
 NOT (1,2,3)
 ESP (2,5,6)
 MTV (1,1,
2)
7. Now for each triangle in Exercise 6, compare the sum of the squares of the two shorter lengths
with the square of the longest length as you did in Exercise 2, only this time put the square of the
longest length on the left of your equation.
 CBS (2,3,3)
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 NOT (1,2,3)
(__)2 ? (__)2 + (__)2
(__)2 ? (__)2 + (__)2
___ ___ ____ +____
___ ___ ____ +____
 ESP (2,5,6)
 MTV (1,1, 2 )
(__)2 ? (__)2 + (__)2
(__)2 ? (__)2 + (__)2
___ ___ ____ +____
___ ___ ____ +____
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8. Did all the sets of sides in Exercise 6 form triangles? (Yes No) How can you be sure that three
given sides will combine to make a triangle?
9. Now use your protractor to measure the angles in each of the triangles you constructed in
Exercise 6. Classify your triangles as acute, right or obtuse.
Have a teammate check the accuracy of your measurements. ______
Initials
 CBS (2,3,3)
 NOT (1,2,3)
 ESP (2,5,6)
 MTV (1,1, 2 )
___________
____________
___________
_____________
10. Compare your findings in Exercises 3 and 9. Summarize this lesson by writing 4 possible
conclusions you can make from today's activities.
1. If______________________________________________then___________________
________________________________________________________________________
2. If______________________________________________then___________________
________________________________________________________________________
3. If______________________________________________then___________________
________________________________________________________________________
4. If______________________________________________then___________________
________________________________________________________________________
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Extension: The right triangle side lengths in Exercise 1 are all positive integers.
A set of three positive integers that satisfy the equation c2 = a2 + b2 is called a Pythagorean
Triple. List all the Pythagorean Triples found in this lesson.
Bonus: How many more Pythagorean Triples can you come up with?
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Homework Lesson Three--Converse of the Pythagorean Theorem
___________________________________________
Name
______
Period
__________________
Date
Directions: Graph points P, Q, and R. Connect the points to form  PQR. Use the
distance formula and the converse of the Pythagorean Theorem to show whether  PQR is
right, acute or obtuse.
Distance formula =
1.
P(3,4), Q(5,0), R(6,2)
2.
P(1,2), Q(4,1), R(0,1)
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( x 2  x1 ) 2  ( y 2  y 1 ) 2
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LESSON PLAN
Lesson Four – &quot;They're SPECIAL!&quot;
Lesson Summary: This lesson is the second part of a two-tiered activity designed to help
students learn right triangle Geometry. Students will continue their study of right triangles as
they use Geometer's Sketchpad to explore 45-45-90 and 30-60-90 triangles of various lengths.
They will then conjecture about the relationships between the legs and the sides of each. They
will chart their findings and then generalize them for each of the special triangles. Finally, they
will complete a chart using only the variables and no specific number values.
Background Knowledge: Students will need to be familiar with: Properties of right triangles,
applying the Pythagorean Theorem, finding squares and square roots in both decimal and radical
form. They also will need a working knowledge of the measuring tools in Geometer's Sketchpad.
Ohio Standards Addressed:
Benchmarks- Number Sense 8-10 H: Find the square root of perfect squares, and approximate
the square root of non-perfect squares.
Benchmarks- Geometry and Spatial Sense 8-10 E: Draw and construct representations of twoand three-dimensional geometric objects using a variety of tools, such as a straight edge,
compass and technology.
Benchmarks-Patterns, Functions and Algebra 8-10 D: Use algebraic representations such as
tables, graphs, expressions, functions and inequalities to model and solve problem situations.
Learning Objectives: Upon completion of this lesson, students will demonstrate knowledge and
be able to:
 State and apply the Pythagorean Theorem, its converse
 Recognize Pythagorean triples and their multiples in right triangle problems
 Determine the lengths of two sides of a 45-45-90 or a 30-60-90 triangle when the length
of a third side is given.
Materials Needed: Pencil, paper, eraser, and handout.
Technology Needed: Individual computers with Geometer's Sketchpad software.
Suggested Procedures:
 Requires advanced preparation: Individual student copies of packet for Activity Four,
determination of grouping for students (1 to 2 per computer).
 &quot;Attention Grabber&quot;: Choose your tool QUIZ. As students enter the room ask them to
choose between three different tools laid out on a table; e.g. Protractor, Compass, Ruler.
Each tool signifies a different problem to solve for a quiz grade.
 Transition: refer to the baseball diamond from last night's homework. Ask the students to
guess at the measure of the angles formed when a diagonal is drawn from second base to
homeplate. Have them verify their guesses. Why do they think it is 45? Are the angle
measurements the same for the angles formed by third and second? Verify. Why is this
so?
 GO TO THE COMPUTER LAB (If you have not taken the students to the lab before,
make sure to explain the rules of the lab and consequences for non adherence.
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Dr. Antonio Quesada - Director, Project AMP 28
Name: __________________
Date: __________________
Period: __________________
Lesson Four: They're Special
Directions: Follow the step by step instructions on this page. Place your answers to all questions
directly on this activity sheet.
1) Double click on the Geometer's Sketchpad icon on your desktop. Open the file imspecial.gsp.
2) Measure all the acute angles of triangles ONE, DAY, GEO, and LAB and identify them in the
space provided. Ex: m&lt;TWO = 87 o
3) What do you notice about the acute angles in each of these triangles?
4) What special name can be given to all of these triangles based on the findings in Ex.2?
5) What special relationship exists between the acute angles in each triangle? Hint: What is their
sum? Give that relationship a name.
6) Measure the larger angle in each of the triangles? Use the proper notation as in Exercise 2.
7) What do you notice about these angles?
8) Based on your answers to Ex. 4 and Ex. 7, write a three word description of all these figures.
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9) Measure the sides of all four triangles. Place their measurements in the Chart 1. (Please read
10 and 11 before completing the 4th column).
45-45-90
Leg 1 Leg 2 Hypotenuse Ratio of legs
to hypotenuse
Triangle ONE
/
Triangle DAY
/
Triangle GEO
/
Triangle LAB
/
10) Using the calculator on your computer, divide the length of the longest leg in each triangle
by the length of one of its shorter legs. Place your answers in the fourth column of the chart in
Exercise 9.
11) What number's square root is this ratio closest to? Place this
chart after the forward slash.
in the fourth column of your
12) Based on your findings how can you find the length (without measuring) of the hypotenuse
of an isosceles right triangle if you know the length of a leg?
13) Based on your findings, how can you find the length of a leg (without measuring) of an
isosceles right triangle if you know the length of the hypotenuse?
14) State your conjectures for exercises 12 and 13 in the form of conditional statements.
&quot;If…then&quot; or &quot;…implies…&quot;
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Dr. Antonio Quesada - Director, Project AMP 30
15) Suppose you label the legs of an isosceles right triangle a and b so that a = b, and the
hypotenuse = c. Is it still true that a2 + b2 = c2 ? Give an example that shows your answer is
true.
16) Using the same variables as in Exercise 15 and the square root you found in Exercise 11,
complete the table below by finding the missing sides in terms of the variable given. The first is
given for you.
Side given
45-45-90
one leg
Leg 1
Leg 2
a
a
one leg
Hypotenuse
a 2
b
hypotenuse _____ ______
c
17) Close this file and open the file urspecial.gsp.
Now (in exercises 18-24) we will repeat exercises 2-9 for this set of triangles. Your
answers to Ex 25 will go into Chart 3.
18) Measure all the acute angles of triangles ONE, DAY, GEO, and LAB and identify them in
the space provided. Ex: m &lt; TWO = 87 degrees.
19) What do you notice about the acute angles in each of these triangles?
20) What special name can be given to all of these triangles based on the findings in Ex.19?
21) What special relationship exists between the acute angles in each triangle? Hint: what is their
sum? Give that relationship a name.
22) Measure the larger angle in each of the triangles? Use the proper notation as in Exercise 18.
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23) What do you notice about these angles?
24) Based on your answers to Ex. 20 and Ex. 23, write a three word description of all these
figures.
25) Measure the sides of all four triangles. Place their measurements in the Chart 3. Again, read
questions 26-28 before filling in column 4 and 5.
Shorter Longer Hypotenuse Ratio of
Ratio of
30-60 Leg
Leg
hypotenuse longer
right
to shorter
leg to
leg
shorter
ONE
/
DAY
/
LAB
/
26) Using the calculator on your computer, divide the length of the hypotenuse in each triangle
by the length of its shorter leg. Place your answers in the fourth column of Chart 3.
27) Using the calculator on your computer, divide the length of the longer leg in each triangle by
the length of its shorter leg. Place your decimal answers in the fifth column of Chart 3.
28) What number's square root is this ratio closest to? Place this
the fifth column of Chart 3.
after the forward slash in
29) Based on your findings how can you find the length of the hypotenuse (without measuring)
of a 30-60-90 triangle if you know… the length of the shorter leg?...the length of the longer leg?
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30) Based on your findings, how can you find the length of the shorter leg (without measuring)
of 30-60-90 triangle if you know the length of the hypotenuse? How about if you know the
length of the longer leg?
31) Based on your findings, how can you find the length of the longer leg (without measuring) of
30-60-90 triangle if you know the length of the hypotenuse? How about if you know the length
of the shorter leg?
32) State your conjectures for exercises 29, 30, and 31 in the form of conditional statements.
&quot;If…then&quot; or &quot;…implies…&quot;
33) Suppose you label the legs of a 30-60-90 triangle a and b so that a  b, and the hypotenuse =
c. Is it still true that a2 + b2 = c2 ? Give an example that shows your answer is true.
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34) Using the same variables as in Exercise 33 and the square root you found in Exercise 28, fill
in the missing cells in Chart 4.
Side given
30-60--90
one leg
one leg
hypotenuse
Leg 1
Leg 2
Hypotenuse
a
a 3
2a
b
_____ ______
c
35) Write your name, date, and period on your paper and turn it in.
Bonus: Can you find a Pythagorean Triple for… an isosceles right triangle?...a 30-60-90 right
triangle? Explain your answers.
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Student imspecial.gsp f ile f or Lesson 4
O
A
L
A
G
E
N
B
E
D
Y
O
A
O
N
L
B
G
E
A
E
Y
O
D
Student urspecial.gsp for Lesson 4
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Name: __________________
Special Right Triangles Homework Worksheet
Exercises 1-6 refer to the 30-60-90 triangle, pictured below.
Applying the general ratios you discovered in your investigation, find the indicated length.
1. AB=14; BC=
4. AB=16; AC=
2. BC=7; AB=
5. AC= 9 3 ; BC=
3. BC=8; AC=
6. AC= 4 3 ; AB=
Exercises 7-12 refer to the 45-45-90 triangle, pictured below.
Applying the general ratios you discovered in your investigation, find the indicated length.
7. XY=7; XZ=
10. XZ=10 ; XY=
8. YZ=10; XZ=
11. YZ= 7 2 ; XZ=
9. XZ= 11 2 ; YZ=
12. XZ=12; YZ=
13. The length of the hypotenuse of a 30-60-90 triangle is 20. What is the length of the shorter
leg?
Bonus: A hexagonal window consists of six congruent panels of glass. Each panel is an
equilateral triangle. Find the area of the entire window if the diagonal that divides the window
into two equal halves is 8 feet long.
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Dr. Antonio Quesada - Director, Project AMP 36
Lesson Five:
Investigation of trigonometric ratios 1
Lesson Summary:
In this lesson we will derive trigonometric ratios for sine, cosine and tangent. Definitions
will be given for cosecant, secant and cotangent.
Key Words:
Trigonometric ratios.
Background Knowledge:
Students will need to have an understanding of right triangles. Some discussion on how
to evaluate sine, cosine and tangent of an angle with a calculator should be given
before they begin. Students must know how to either use a protractor and a ruler or
how to operate the Geometer’s sketchpad (it is up to the teacher whether or not to use
the sketchpad). In addition, similar triangles must have been studied before the
extension can be answered.
NCTM Standard(s) Addressed:
1) This lesson addresses the geometry standard for 9-12.
Specifically the appropriate benchmark is as follows:
“Analyze characteristics and properties of two- and three-dimensional geometric
shapes and develop mathematical arguments about geometric relationships”.
The appropriate objectives for this benchmark are:
 Establish the validity of geometric conjectures using deduction, prove theorems,
and critique arguments made by others;
 Use trigonometric relationships to determine lengths and angle measures.
2) This lesson also addressed the problem solving standard for 9-12.
Specifically the appropriate objectives are as follows:
“Instructional programs from prekindergarten through grade 12 should enable all
students to build new mathematical knowledge through problem solving”.
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Dr. Antonio Quesada - Director, Project AMP 37
3) This lesson also addresses the communication standard for 9-12.
Specifically the appropriate objectives are as follows:
“Instructional programs from prekindergarten through grade 12 should enable
all students to—




organize and consolidate their mathematical thinking through communication;
communicate their mathematical thinking coherently and clearly to peers,
teachers, and others;
analyze and evaluate the mathematical thinking and strategies of others;
Use the language of mathematics to express mathematical ideas precisely.
Learning Objectives (as stated from NCTM):
Establish the validity of geometric conjectures using deduction, prove theorems, and
critique arguments made by others by developing formulas for trigonometric ratios
(NCTM).
Use trigonometric relationships to determine lengths and angle measures (NCTM).
Students will build new mathematical knowledge through problem solving (NCTM).
Students will organize and consolidate their mathematical thinking through
communication (NCTM).
Students will be able to communicate their mathematical thinking coherently and clearly
to peers, teachers, and others (NCTM).
Students will analyze and evaluate the mathematical thinking and strategies of others
(NCTM).
Students will use the language of mathematics to express mathematical ideas precisely
(NCTM).
In addition, students will make connections to previously studied similar triangles.
Materials:
1) Handout of Trigonometric ratios.
2) A calculator capable of evaluating sine, cosine and tangent (be sure it is in degree
mode).
3) Graph paper, protractor and ruler or Sketchpad (and knowledge of how to use the
Sketchpad).
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Dr. Antonio Quesada - Director, Project AMP 38
Suggested Procedures:
Be sure to discuss degree mode and how to evaluate sine, cosine and tangent of an
angle with the students.
Let them know there is a relationship between ratios of a right triangle and the sine,
cosine and tangent of angles of a right triangle.
Students should be grouped in groups of three if they are going to use graph paper,
protractors and a ruler. We would consider grouping them in pairs if they are using the
Sketchpad because it may be too crowded at the computers. You may want to assign
one group member the job of recording, one the job of calculating and one the job of
checking. We personally believe all group members should be calculating, double
checking and discussing.
Extensions 1-3 could be given as homework if there is not enough time in class for a
wrap up discussion of the activity. This discussion could continue the following day
after the extensions are finished.
Assessment:
Communication in groups:
a) The first form of assessment would be done while the groups are working on the
investigation.
Communication to the class:
a) An assessment can be made during the follow up discussion of the activity. Each
group could take a turn presenting their findings to the class at random. The group
should be at an agreement upon their findings.
Homework:
a) It may be helpful to require that every student submit the extensions to the activity.
This can be collected and graded, discussed or simply spot checked.
b) Additional homework problems can be assigned. See additional homework problems
and solutions.
Quiz/Test:
a) A quiz or test over this material or the unit as a whole can be given.
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Dr. Antonio Quesada - Director, Project AMP 39
Name: __________________
Investigation of trigonometric ratios
In this lesson we will derive trigonometric ratios for sine, cosine and tangent. Definitions will be
given for cosecant, secant and cotangent.
Definition:
If we consider angle ABC (labeled  ), we can define the hypotenuse to be side BC, the
adjacent side (next to angle  ) to be side AB and the opposite side (opposite the angle  ) to be
side AC.
Opposite 
AC = 16.41 cm
A
mBCA = 21.00
C
mBAC = 90.00
BA = 6.30 cm
Adjacent to 
BC = 17.57 cm
mABC = 69.00
Hypotenuse

1) Please calculate the following ratios for the right triangle above. Write your answer as a
decimal rounded to four decimal places.
a)
Opposite
AC


Hypotenuse BC
b)
Adjacent

Hypotenuse
c)
Opposite

Adjacent
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Dr. Antonio Quesada - Director, Project AMP 40
2) Using your calculator and the right triangle above, please find the following Trigonometric
functions below. Be sure your calculator is in degree mode (press the “mode” key then go
over to “degree” instead of “radian”).
a) Sin (  ) =
b) Cosine (  ) 
c) Tangent (  ) 
3) Do you notice anything about your calculations for question 1 and 2? Explain in full detail
the relationships that you found.
4) On graph paper (or with sketchpad), create your own right triangle and see if the
relationships that you found in question 3 are still true.
5) Were the relationships discovered in question 3 still true. Why or why not? Please explain
with calculations.
6) Please write a general rule for the relationships that you found.
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Definition:
You have discovered SOHCAHTOA or
opposite
adjacent
opposite
.
sin  
, cos 
, and tan  
hypotenuse
hypotenuse
adjacent
The trig functions above are abbreviated but are read as “sine, cosine and tangent”,
respectively.
There are three additional trigonometric functions “cosecant, secant and cotangent” that are
found by inverting each of the three above functions. Therefore, we also have that
1
hypotenuse
1
hypotenuse
1
adjacent
csc  

, sec  

, and cot  

.
sin 
opposite
cos 
adjacent
tan  opposite
7) Please summarize in detail what you learned in this lesson.
Extension 1:
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Using the original triangle (from page 1), set angle BCA =  , then find the trigonometric
relations for  . Don’t forget to rename your opposite and adjacent sides based on  .
Extension 2:
Now compare the trigonometric ratios for  and  . (Remember that we found these in
Extension 1 and in problem 3). Do you notice any connection between these trigonometric
ratios?
Extension 3:
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Dr. Antonio Quesada - Director, Project AMP 43
Suppose we had another right triangle as shown below. Notice this triangle also has angles
measuring 90, 21 and 69 degrees.
Opposite 
A
BA = 3.36 cm
Adjacent to 
AC = 8.75 cm
mBCA = 21.00
C
mBAC = 90.00
mABC = 69.00

BC = 9.38 cm
Hypotenuse
B
Question:
Set up three trigonometric ratios using  involving all three sides of the triangle above.
a) Do you notice any similarities in the calculations for this triangle and the triangle
given in exercise 1?
b) What are the similarities specifically? Please explain.
c) Why do you think this would happen (think back)?
Name: ________________________
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Dr. Antonio Quesada - Director, Project AMP 44
Homework for Trigonometric Ratios
1) Explain why the two triangles below are similar.
55

55

What else will be the same in each triangle?
2) Which of the four statements below is/are true about ABC?
a
b
c
b) cos A 
b
c
c) tan B 
a
a
d) sin A 
c
a) sin C 
a
C
B
c
b
A
3) Given sin 20 o  0.342 find the value of x in the diagram, correct to 1 decimal place.
8
x
20
4) Find the value of k, correct to 1 decimal place. Show all work.
9.5
72
k
5) An escalator at an airport slopes at an angle of 30&deg; and is 20 m long. Through what height
would a person be lifted by travelling on the escalator?
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6) The top of a flagpole is connected to the ground by a cable 12 meters long. The angle that the
cable makes with the ground is 40.
Find the height of the flagpole.
40
7) A ship’s navigator observes a lighthouse on a cliff. She knows from a chart that the top of the
lighthouse is 35.7 meters above sea level. She measures the angle of elevation of the top of
the lighthouse to be 0.7.
35.7
m
0.7
The coast is very dangerous in this area and ships have been advised to keep at least 4 km from
this cliff to be safe.
Is the ship safe?
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Lesson Six
Investigation of Inverse Trig functions
Lesson Summary:
This investigation will rely on previously discovered trigonometric ratios and will focus on
leading students to understand how to use the inverse of trigonometric functions to find angles of
a right triangle. Practical applications involving real world connections will also be studied.
Key Words:
Finding angles with trig ratios.
Background Knowledge:
Students will need to have an understanding of right triangles and the Pythagorean
Theorem. Some discussion on how to evaluate sin-1, cos-1 and tan-1 of an angle with a
calculator should be given before they begin. This lesson assumes the previous
discussion of trigonometric ratios.
NCTM Standard(s) Addressed:
1) This lesson addresses the geometry standard for 9-12.
Specifically the appropriate benchmark is as follows:
“Analyze characteristics and properties of two- and three-dimensional geometric
shapes and develop mathematical arguments about geometric relationships”.
The appropriate objectives for this benchmark are:
 Establish the validity of geometric conjectures using deduction, prove theorems,
and critique arguments made by others;
 Use trigonometric relationships to determine lengths and angle measures
2) This lesson also addressed the problem solving standard for 9-12.
Specifically the appropriate objectives are as follows:
“Instructional programs from prekindergarten through grade 12 should enable all
students to build new mathematical knowledge through problem solving”.
Project AMP
Dr. Antonio Quesada - Director, Project AMP 47
3) This lesson also addresses the communication standard for 9-12.
Specifically the appropriate objectives are as follows:
“Instructional programs from prekindergarten through grade 12 should enable
all students to—




organize and consolidate their mathematical thinking through communication;
communicate their mathematical thinking coherently and clearly to peers,
teachers, and others;
analyze and evaluate the mathematical thinking and strategies of others;
Use the language of mathematics to express mathematical ideas precisely.
4) This lesson also addresses the connection standard 9-12.
Specifically the appropriate objectives are as follows:
“Instructional programs from prekindergarten through grade 12 should enable
all students to—



recognize and use connections among mathematical ideas;
understand how mathematical ideas interconnect and build on one another to
produce a coherent whole;
Recognize and apply mathematics in contexts outside of mathematics.
Learning Objectives (as stated from NCTM):
Establish the validity of geometric conjectures using deduction, prove theorems, and
critique arguments made by others (NCTM).
Use trigonometric relationships to determine lengths and angle measures (NCTM).
Students will build new mathematical knowledge through problem solving (NCTM).
Students will organize and consolidate their mathematical thinking through
communication (NCTM).
Students will be able to communicate their mathematical thinking coherently and clearly
to peers, teachers, and others (NCTM).
Students will analyze and evaluate the mathematical thinking and strategies of others
(NCTM).
Students will use the language of mathematics to express mathematical ideas precisely
(NCTM).
Students will make connections to the previously studied Pythagorean Theorem.
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Students will recognize and use connections among mathematical ideas.
Students will understand how mathematical ideas interconnect and build on one another
to produce a coherent whole.
Students will recognize and apply mathematics in contexts outside of mathematics.
Materials:
1) Handout of Inverse Trig functions.
2) A calculator capable of evaluating sin-1, cos-1 and tan-1 (be sure in degree mode).
Suggested Procedures:
Be sure to discuss degree mode and how to evaluate sin -1, cos-1 and tan-1 of an angle
with the students.
Let them know that they will be using the previously discovered relationship between
ratios of a right triangle and the sine, cosine and tangent of angles of a right triangle.
Students should be grouped in groups of three. You may want to assign one group
member the job of recording, one the job of calculating and one the job of checking. We
personally believe all group members should be calculating, double checking and
discussing.
Extensions 1-2 could be given as homework if there is not enough time in class for a
wrap up discussion of the activity. This discussion could continue the following day
after the extensions are finished.
Assessment:
Communication in groups:
a) The first form of assessment would be done while the groups are working on the
investigation.
Communication to the class:
a) An assessment can be made during the follow up discussion of the activity. Each
group could take a turn presenting their findings to the class at random. The group
should be at an agreement upon their findings.
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Dr. Antonio Quesada - Director, Project AMP 49
Homework:
a) It may be helpful to require that every student submit the extensions to the activity.
This can be collected and graded, discussed or simply spot checked.
b) Additional homework problems can be assigned.
problems.
See additional homework
Quiz/Test
a) A quiz or test over this material or the unit as a whole can be given.
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Dr. Antonio Quesada - Director, Project AMP 50
Name: ________________
Investigation of Inverse Trig functions
This investigation will rely on previously discovered trigonometric ratios and will focus on
leading students to understand how to use the inverse of trigonometric functions to find angles of
a right triangle. Practical applications will also be studied.
Suppose we do not know the measure of angle A (labeled  ) or the measure of angle C (labeled
 ) in the triangle below. We want to be able to find these angles but we only know the lengths
of the sides of the right triangle. Using SOHCAHTOA (as was previously discovered), we will
find the missing angles.
BC = 6.27 cm

B
mABC = 90.00
C
BA = 4.82 cm

AC = 7.90 cm
A
Definition:
We need to know that given sin A 
a
a
, we can find A by taking sin 1  . This means that
b
b
a
sin 1    A. This definition will also be true for any of the other trigonometric functions.
b
1) By SOHCAHTOA, we know that sin  
BC 6.27

 0.79.
AC 7.90
a) Find  using the definition above.
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b) Use the information given in the right triangle above to set up a similar equation for
cos ine to find angle  .
c) Does this also give you the same value for angle  ?
d) If the tangent function is used to find angle  , do you think you will find the same
number for  as we previously found? Show the details of your calculation for the
tangent function below.
2) Now use sine, cosine or tangent, to find the measure of angle  .
3) What if I had a right triangle but only knew two (of the three) side lengths. For example,
in the triangle above, suppose I know that angle  is 90 degrees, side length BC = 6.27
and side AC = 7.90. Do I need to know the length of side AB or can I find the length of
AB somehow? Please explain below.
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4) Summarize what you learned in this lesson.
Extension 1:
An airplane flying at an altitude of 30,000 feet is headed toward an airport. To guide the
airplane to a safe landing, the airport’s landing system sends radar signals from the runway to
the airplane at a 10 degree angle of elevation. How far is the airplane (measured along the
ground) from the airport runway? Hint: Set up a trigonometric equation (using
SOHCAHTOA) and solve for the unknown variable. (Be sure you are in degree mode on
your calculator. Type mode, then go over to degree from radians).
A
30,000 ft.
10 degrees
B
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Dr. Antonio Quesada - Director, Project AMP 53
Extension 2:
You are standing 75 meters from the base of the Jin Mao Building in Shanghai, China. You
estimate that the angle of elevation to the top of the building is 80 degrees. What is the
approximate height of the building? Suppose one of your friends is at the top of the building.
What is the distance between you and your friend? (Be sure you are in degree mode on your
calculator. Type mode, then go over to degree from radians).
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Dr. Antonio Quesada - Director, Project AMP 54
Name: ________________________
Homework for Trigonometric Ratios
1) I used a calculator and found the tan ratio of a certain angle to be 1.234. What could the
size of the angle be?
2) Toni wants to find the value of  in the triangle below. Which statement is correct?
3
a) cos   
5
b) sin  
c) tan  
5
4
4
5

3
3
4
d) sin 0.8  
3) Find out everything you can about the right-angled triangle below.
35 cm
43
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4) In the examples below, you are asked to demonstrate that you can do something, by doing
it. In each case, you get to pick the examples that you think will show that you can do what
has been asked.
a) I can understand and use the trigonometric ratios (sine, cosine, tangent) in right-angled
triangles:
b) I can use trigonometry to solve practical problems involving right-angled triangles:
5) A crane has a 200-foot arm whose lower end is 5 feet off the ground. The arm has to reach
the top of the dome 80 feet high. At what angle X should the arm be set?
Hint: We must adjust for the 5 feet off the ground that the crane arm sits by allowing the other
side to measure 75 feet high.
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```