student workbook with scaffolded practice unit 3 · 2017-08-09 · unit 3 • trigonometry of...

Student Workbookwith Scaffolded Practice

Unit 3

1

1 2 3 4 5 6 7 8 9 10

ISBN 978-0-8251-7456-8 U3

Copyright © 2014

J. Weston Walch, Publisher

Portland, ME 04103

www.walch.com

Printed in the United States of America

EDUCATIONWALCH

This book is licensed for a single student’s use only. The reproduction of any part, for any purpose, is strictly prohibited.

© Common Core State Standards. Copyright 2010. National Governor’s Association Center for Best Practices and

Council of Chief State School Officers. All rights reserved.

2

Program pages

Workbook pages

Introduction 5

Unit 3: Trigonometry of General Triangles and Trigonometric FunctionsLesson 1: Radians and the Unit Circle

Lesson 3.1.1: Radians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .U3-5–U3-21 7–16

Lesson 3.1.2: The Unit Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .U3-22–U3-45 17–28

Lesson 3.1.3: Special Angles in the Unit Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . .U3-46–U3-69 29–38

Lesson 3.1.4: Evaluating Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . .U3-70–U3-92 39–48

Lesson 2: Trigonometry of General AnglesLesson 3.2.1: Proving the Law of Sines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U3-101–U3-124 49–62

Lesson 3.2.2: Proving the Law of Cosines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U3-125–U3-143 63–74

Lesson 3.2.3: Applying the Laws of Sines and Cosines . . . . . . . . . . . . . . . . . . . U3-144–U3-161 75–84

Lesson 3: Graphs of Trigonometric FunctionsLesson 3.3.1: Periodic Phenomena and Amplitude, Frequency,

and Midline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U3-169–U3-186 85–94

Lesson 3.3.2: Using Trigonometric Functions to Model Periodic Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U3-187–U3-205 95–106

Station ActivitiesSet 1: Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U3-228–U3-237 107–116

Set 2: The Laws of Sines and Cosines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U3-243–U3-250 117–124

Coordinate Planes 125–142

Table of Contents

CCSS IP Math III Teacher Resource© Walch Educationiii

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The CCSS Mathematics III Student Workbook with Scaffolded Practice includes all of the student pages from the Teacher Resource necessary for your day-to-day classroom use. This includes:

• Warm-Ups

• Problem-Based Tasks

• Practice Problems

• Station Activity Worksheets

In addition, it provides Scaffolded Guided Practice examples that parallel the examples in the TRB and SRB. This supports:

• Taking notes during class

• Working problems for preview or additional practice

The workbook includes the first Guided Practice example with step-by-step prompts for solving, and the remaining Guided Practice examples without prompts. Sections for you to take notes are provided at the end of each sub-lesson. Additionally, blank coordinate planes are included at the end of the full unit, should you need to graph.

The workbook is printed on perforated paper so you can submit your assignments and three-hole punched to let you store it in a binder.

CCSS IP Math III Teacher Resource© Walch Educationv

Introduction

5

UNIT 3 • TRIGONOMETRY OF GENERAL TRIANGLES AND TRIGONOMETRIC FUNCTIONSLesson 1: Radians and the Unit Circle

U3-5

Name: Date:

CCSS IP Math III Teacher Resource 3.1.1

© Walch Education

Warm-Up 3.1.1

Marissa is running a race on a circular track that has a radius of 200 feet. She is competing against three other runners, and they have all taken their marks along the inside of the track. Marissa’s coach stands in the center of the circle to watch as the race gets underway. After 30 seconds, the coach notes each racer’s location:

• Marissa is about 125° from the starting point.

• The second racer is about 115° from the starting point.

• The third racer is about 120° from the starting point.

• The fourth racer is about 135° from the starting point.

For any circle, πθ

=s r180

, where s = arc length, r = radius, and θ = the measure of the central angle

of the arc in degrees. Use this information to answer the questions that follow. Round your answers to

the nearest foot.

1. Approximately how far has Marissa run around the track?

2. Approximately how far has the second racer run?

3. Approximately how far has the third racer run?

4. Approximately how far has the fourth racer run?

5. Approximately how far is Marissa from the leader?

Lesson 3.1.1: Radians

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U3-12CCSS IP Math III Teacher Resource 3.1.1

© Walch Education

Name: Date:

Scaffolded Practice 3.1.1Example 1

Given the diagram of B , find the measure of θ in radians. Round your answer to the nearest ten-thousandth.

AC = 19 in7 in

θ

B

C

A

1. Identify the length of the radius and the arc length.

2. Substitute r and s into the formula θ =s

r and solve for θ.

continued

9


U3-13CCSS IP Math III Teacher Resource

3.1.1© Walch Education

Name: Date:

Example 2

Convert 78° to radians. Give your answer as an exact answer and also as a decimal rounded to the nearest ten-thousandth.

Example 3

Convert π2

3 radians to degrees.

Example 4

Convert 0.5793 radian to degrees. Round your answer to the nearest tenth.

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Problem-Based Task 3.1.1: A Satellite’s Story

A communications satellite located approximately 22,223 miles above the Earth’s surface orbits the planet at a speed of about 7,000 miles per hour. What is the satellite’s rotational speed in radians per hour and in degrees per hour? The diameter of the Earth is 7,918 miles. Round radians to the nearest thousandth and degrees to the nearest tenth.

What is the satellite’s

rotational speed in radians per hour and in degrees per

hour?

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U3-20

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© Walch Education

For problems 1–3, use the information in the diagrams to find the angle measure of θ in radians. Round your answer to the nearest ten-thousandth, if necessary.

1. YZ = 17 in

4 in

θ

XY

Z

2. BD = 12 ft

5 ft

θ

CD

B

3. RS = 8.5 mm

13.6 mm

θ

O

S

R

Practice 3.1.1: Radians

continued

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U3-21

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© Walch Education

For problems 4–7, convert each radian measure to degrees. Round your answer to the nearest tenth, if necessary.

4. π8

9 radians

5. π15

11 radians

6. 3.0672 radians

7. 0.9431 radian

Read the following scenario, and use the information in it to complete problems 8–10.

Mitra, Charlene, and Abbie are taking turns spinning the merry-go-round at the park. They each spin the carousel at a different speed in degrees per second. How fast is each girl’s spin in radians per second? Supply an exact answer and also a decimal approximation rounded to the nearest ten-thousandth.

8. Mitra’s spin speed: 215° per second

9. Charlene’s spin speed: 21° per second

10. Abbie’s spin speed: 104° per second

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Notes

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Notes

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U3-22

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© Walch Education

Warm-Up 3.1.2

Anna and her friend Isabella are sitting on a spinning ride at their favorite amusement park. Once

the ride gets up to speed, it spins at a rate of 115° per second. Use this information to answer the

questions that follow. Remember that θ =s

r.

1. How fast does the ride spin in radians per second? Give an exact answer.

2. Anna is sitting 22 feet from the center of the ride. How far does she travel each second? Round your answer to the nearest tenth.

3. Isabella is sitting 17 feet from the center of the ride. How far does she travel each second? Round your answer to the nearest tenth.

Lesson 3.1.2: The Unit Circle

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On a unit circle, sketch angles that measure π2

3 radians,

π4

radian, and π9

7 radians.

1. Sketch a unit circle, and then label π radians and 2π radians.

2. Sketch π2

3 radians.

3. Sketch π4

radian.

4. Sketch π9

7 radians.

5. Summarize your findings.

continued

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Example 2

Find the reference angles for angles that measure π11

9 radians,

π3

5 radians, and 5.895 radians.

Example 3

Use the following diagram of an angle in the unit circle to demonstrate why the point where the terminal side intersects the unit circle is (cos θ, sin θ).

θ x

y

continued

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Example 4

Find the coordinates of the point where the terminal side intersects the unit circle. Round each coordinate to the nearest hundredth.

x

y

θ =6π

7radians 0 radians

2π radiansπ radians

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Problem-Based Task 3.1.2: A Sticker’s Journey

While riding home from school one day, Darius pauses to show his friend Emma his new bicycle. He

explains that the wheels each have a radius of 1 foot. Emma notices that the front tire has a sticker on

it, located exactly to the right of the center of the wheel. When Darius backpedals, the sticker moves

counterclockwise around the wheel. Emma observes that after Darius backpedaled for 1 second,

the sticker had moved 2.5 feet from its original location. At that point, how far is the sticker from

the ground? Round your answer to the nearest tenth. Recall the formula θ =s

r, where θ is the angle

measure in radians, s is the arc length, and r is the radius.

How far is the sticker from

the ground?

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U3-45

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© Walch Education

For problems 1–3, sketch each radian measure on the unit circle.

1. π2

9 radians

2. π8

5 radians

3. π5

6 radians

For problems 4–7, find the reference angle for each angle measure.

4. 318°

5. π4

5 radians

6. π3

radians

7. 4.087 radians

For problems 8–10, find the coordinates of the point where the terminal side of the angle intersects the unit circle. Round each coordinate to the nearest hundredth.

8. π5

4 radians

9. π6

radian

10. 2.834 radians

Practice 3.1.2: The Unit Circle

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Notes

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U3-46

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© Walch Education

Warm-Up 3.1.3

Raeann and her three friends are all in the school’s marching band. For their new halftime show, one of their formations is an equilateral triangle. Raeann, Chris, and Judson stand at the three vertices of the triangle (points R, C, and J, respectively). Their friend Trisha stands halfway between Chris and Judson (at point T ). Raeann and Chris are 20 feet apart. Use this information and the diagram to answer the questions that follow.

C

R

JT

1. How far apart are Chris and Judson?

2. How far apart are Chris and Trisha?

3. How far apart are Raeann and Trisha? Give an exact answer.

4. What are the measures of the angles of CTR in degrees?

Lesson 3.1.3: Special Angles in the Unit Circle

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Find the coordinates of the point where the terminal side of a 330° angle intersects the unit circle.

1. Sketch the angle on the unit circle and identify the location of the terminal side.

2. Identify the reference angle.

3. Find the cosine and sine of the reference angle.

4. Determine the coordinates of the point where the terminal side intersects the unit circle.

continued

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© Walch Education

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Example 2

Find the coordinates of the point where the terminal side of an angle with a measure of π5

4 radians

intersects the unit circle.

Example 3

Find the coordinates of the point where the terminal side of an angle with a measure of π3

2 radians

intersects the unit circle.

Example 4

Sketch the three special angles that are located in Quadrant II. Label the coordinates of the points where their terminal sides intersect the unit circle. Use degrees.

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Problem-Based Task 3.1.3: Ferris Wheel Fun

Andre is waiting for his friends to finish riding a Ferris wheel that has a radius of 1 decameter (1 decameter = 10 meters). When Andre first looks over at the wheel, he sees that Veronica and Noelle are in a seat that is directly to the right of the center of the wheel. His other friends, Reid and Kip, are in a seat that is exactly at the top of the Ferris wheel. A minute later, he sees that the wheel has rotated exactly 120° counterclockwise as additional passengers are loaded. At this point, what is the distance from Reid and Kip’s seat to Veronica and Noelle’s seat? Give an exact answer.

What is the distance from

Reid and Kip’s seat to Veronica and Noelle’s seat?

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U3-69

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© Walch Education

For problems 1–9, find the coordinates of the point where the terminal side of the angle intersects the unit circle. Give exact answers.

1. 45°

2. 210°

3. 90°

4. 300°

5. π3

radians

6. π7

4 radians

7. 2π radians

8. π5

6 radians

9. π7

6 radians

Use your knowledge of unit circles to complete problem 10.

10. Create a unit circle that contains all the special angles in degrees. Label the terminal point of each angle with its coordinates.

Practice 3.1.3: Special Angles in the Unit Circle

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Notes

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U3-70

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© Walch Education

Warm-Up 3.1.4

A group of firefighters is called in to rescue a kitten stuck in a tree. The firefighters lean a 25-foot ladder up against the tree so that the base of the ladder rests 6 feet away from the base of the tree, as shown in the diagram. Answer the following questions and round each answer to the nearest tenth.

LadderTree

C A

B

1. How high up the tree does the ladder reach?

2. What is the measure of the angle the ladder makes with the ground (angle A)?

3. As the firefighter climbs down the ladder with the kitten, he stops 3 feet from the base of the ladder and jumps off. How far off the ground was he when he jumped?

Lesson 3.1.4: Evaluating Trigonometric Functions

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What is the sign of each trigonometric ratio for an angle with a measure of π9

8 radians?

1. Sketch the angle to determine in which quadrant it is located.

2. Determine the signs of the lengths of the opposite side, adjacent side, and hypotenuse for the reference angle.

3. Use the definitions of the trigonometric functions to determine the sign of each and check the results by using the acronym ASTC.

continued

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© Walch Education

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Example 2

Find sin θ if θ is a positive angle in standard position with a terminal side that passes through the point (5, –2). Give an exact answer.

Example 3

Find π

csc2

3. Give an exact answer.

Example 4

Given θ =cos4

5, if θ is in Quadrant I, find cot θ.

Example 5

Find cos θ if θ is a positive angle in standard position with a terminal side that passes through the point (–1, 0). Give an exact answer.

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Problem-Based Task 3.1.4: Spring into Action

A weight hangs on the end of a coiled spring; when the weight is pulled and released, the spring

begins to bounce up and down at regular intervals. The weight’s displacement, or distance, from

equilibrium (its resting position) is given by the formula =y t t( )1

2cos 3 , in which y(t) is the

displacement in feet and t is the time in seconds. Randall snaps a photo 0.1 second after the weight is

released and again exactly 1 second later. How much higher, to the nearest thousandth, is the weight

in the first photo than in the second photo? (Note that the formula uses radians.)

How much higher, to the nearest

thousandth, is the weight in the first photo than in the second photo?

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U3-92

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© Walch Education

For problems 1–4, determine the specified trigonometric ratio for each angle with a terminal side that passes through the given point. Give exact answers.

1. cos θ ; (–2, 7)

2. cot θ ; (4, –5)

3. sec θ ; (0, –1)

4. sin θ ; (–12, –5)

For problems 5–7, determine the specified trigonometric ratio for each special angle. Give exact answers.

5. π

sin7

6 radians

6. π

tan3

4 radians

7. π

csc3

radians

For problems 8–10, each angle is described by one of its trigonometric ratios and the quadrant in which its terminal side is located. Find the requested trigonometric ratio for the angle. Give an exact answer.

8. Find cos θ given θ =tan2

3 with a terminal side in Quadrant III.

9. Find sec θ given θ =−sin3

5 with a terminal side in Quadrant IV.

10. Find cot θ given θ =−cos1

2 with a terminal side in Quadrant II.

Practice 3.1.4: Evaluating Trigonometric Functions

45

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CCSS IP Math III Teacher Resource3.2.1

© Walch Education

UNIT 3 • TRIGONOMETRY OF GENERAL TRIANGLES AND TRIGONOMETRIC FUNCTIONSLesson 2: Trigonometry of General Angles

U3-101

Name: Date:

Warm-Up 3.2.1

Alexandra planted a vegetable garden and split it into three different plots, as shown in the diagram. Use this diagram to answer the questions that follow. Round answers to the nearest tenth.

C

7 ft

B

A

2 ft

3 ft68˚

1. Alexandra planted herbs in plot A. What is the area of this plot? Hint: To find the height of the

triangle, use the tangent ratio, θ =tanlength of opposite side

length of adjacent side.

2. What is the area of the entire garden?

3. If Alexandra wants to put up a fence around plot A, how much fencing will she need to buy?

Lesson 3.2.1: Proving the Law of Sines

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© Walch Education

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Prove the Law of Sines, = =a

A

b

B

c

Csin sin sin, using the given information for the diagram of ABC

with altitude h and sides a, b, and c.

b

B

A C

hc

a

1. Use the definition of sine to create statements for angles A and C.

2. Solve sin A and sin C for h.

3. Set the equations equal to each other and rearrange.

4. Repeat the proof for angles B and C.

5. Use the definition of sine to create statements for angles B and C.

6. Solve the equations for sin B and sin C for h.

7. Set the equations equal to each other and rearrange.

8. Combine the two resulting formulas from steps 3 and 7 to complete the proof.

continued

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Example 2

For ABC , find the length of side b if π

∠ =m B3

radians and π

∠ =m C2

9 radians.

π39

2

5 cm

B

C

A

b

π33

Example 3

For ABC , find the measure of angle C if ∠ = °m A 34 , c = 9 meters, and a = 7 meters.

continued

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© Walch Education

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Example 4

Derive the formula A ab C=1

2sin for the area of ABC with altitude h.

b

B

A C

hc

a

Example 5

Find the area of DEF using the given information.

22°

12 cm

8 cmD E

F

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© Walch Education


U3-118

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Problem-Based Task 3.2.1: Count the Cost

Mr. Soliz wants to enclose a triangular plot of land to create a pen for his sheep and goats. He estimates that fencing will cost $10 per foot. He also wants to plant a certain type of grass, which he estimates will cost $0.03 per square foot. How much money, to the nearest dollar, will he need for the entire project?

30 ft

65° 33°55 ft

How much money, to the nearest dollar, will he

need for the entire project?

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U3-122

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For problems 1–5, use the Law of Sines to find each requested measurement. Round answers to the nearest hundredth.

1. What is the length of FG ?

9 ft37°

39°

104°

F

G

E

2. If π

∠ =m B2

3 radians, what is the measure of angle A?

8 ft

7 ft

13 ft

B

C

A

3. For UVW , UV = 6 mm, VW = 8 mm, and ∠ = °m W 34 . What is the measure of angle U ?

4. For ABC, BC = 7 inches, ∠ =m A 0.245 radians, and ∠ =m C 0.784 radians. What is the length of AB?

5. For XYZ , XY = 8 feet, XZ = 7 feet, and π

∠ =m Y4

radians. What is the measure of angle Z ?

Practice 3.2.1: Proving the Law of Sines

continued

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© Walch Education


U3-123

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For problems 6 and 7, find the area of each triangle. Round answers to the nearest hundredth.

6. For CDE , CD = 11 meters, DE = 6 meters, EC = 8 meters, and ∠ = °m C 112 .

7. For ABC as shown, π

∠ =m A7

radians.

9 cm

5 cm11 cm

C

B

A

For problems 8–10, find the perimeter of each triangle. Round answers to the nearest tenth.

8.

13.8 km

45°

67°

10.4 km

continued

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© Walch Education


U3-124

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9. 34°

29°

5 mi

10. 0.436 radians

2.409 radians

14 m

59

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© Walch Education


U3-125

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Warm-Up 3.2.2

Paul, Nate, and Kirsten want to practice shooting their paintball guns. They set up a target and stand due south of it. Nate then walks 12 feet to the west to face the target at a 60° angle. Kirsten walks 15 feet to the east to face the target at a 55° angle. The diagram shows the positions of Nate, Paul, and Kirsten (N, P, and K ) in relation to the target, T. Use the diagram to answer the questions that follow. Round answers to the nearest whole foot.

T

KN P60° 55°

15 ft12 ft

1. How far is Nate from the target?

2. How far is Kirsten from the target?

3. How far is Paul from the target?

Lesson 3.2.2: Proving the Law of Cosines

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Given the diagram of ABC with altitude h, prove the Law of Cosines.

B

DC A

b

ca

h

yx

1. Create statements for h, x, and y using the sine and cosine of angle C.

2. Use the Pythagorean Theorem and the known expressions to write an equation for ADB .

3. Expand and rearrange the statement to develop the Law of Cosines.

continued

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Example 2

Given the diagram of DEF , find the length of side f. Round the length to the nearest tenth.

29°

f15 ft

9 ft

D

E F

Example 3

Given the diagram of LMN , find the measure of angle L in radians.

8 ft

5 ft

6 ft

L

M

N

continued

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Example 4

Given the diagram of XYZ , solve the triangle by finding the missing sides and angles.

z

86°

14 m

10 m

Z

Y

X

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© Walch Education


U3-138

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Problem-Based Task 3.2.2: Travel Trouble

An airplane took off from an airport (point A) and flew due west for 1 hour, 50 minutes at an average ground speed of 410 miles per hour (mph) until it experienced technical problems and had to divert to a nearby airport to land. The plane turned 45° to the north and traveled for 15 minutes at an average speed of 285 mph before landing. When the plane landed (at point C ), how far was it from its original departure airport? Use the equation for distance, d = rt, where d is distance, r is speed, and t is time. Round answers to the nearest mile. Assume all distances and speeds include the effects of the wind.

45°A

B

C

When the plane landed (at point C ), how far was

it from its original departure airport?

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U3-142

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For problems 1–4, use the Law of Cosines to find the measure of each requested angle. Round degrees to the nearest tenth and radians to the nearest thousandth.

1. For VWX , find the measure of angle V in degrees if VW = 15 meters, WX = 10 meters, and VX = 9 meters.

2. For NMO , find the measure of angle O in radians if NM = 4 inches, NO = 5 inches, and MO = 8 inches.

3. Find the measure of angle I in radians.

16 cm13 cm

11 cmI

K

J

4. Find the measure of angle B in degrees.

3 in7 in

6 inB

A

C

Practice 3.2.2: Proving the Law of Cosines

continued

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© Walch Education


U3-143

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For problems 5–7, solve each triangle by finding the measure of all the unknown angles and sides. Use degrees for angle measurements and round all answers to the nearest whole number.

5. For RST , RS = 8 in, RT = 5 in, and ∠ = °m R 86 .

6. For ABC , AB = 9 meters, BC = 4 meters, and AC = 6 meters.

7. For HIJ , refer to the diagram.

7 cm

10 cm

123°

I

HJ

For problems 8–10, determine the amount of fencing that would be needed to completely enclose each triangular plot of land. Round answers to the nearest tenth.

8.

34°2.8 mi

4.5 mi

9.

133°4.8 km

5.6 km

10.

3π5

radians66 m49 m

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U3-144

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Warm-Up 3.2.3

Mr. Lawson bought boards to build a triangular-shaped sandpit for his sons. He cut two of the boards to be 4 feet long and the third to be 6 feet long. Answer the following questions about the sandpit. Use degrees for angle measures and round answers to the nearest tenth.

1. What will be the measure of the angle formed by the 4-foot boards?

2. Determine the measures of the other two angles.

3. What will be the area of the sandpit?

Lesson 3.2.3: Applying the Laws of Sines and Cosines

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A surveyor needs to measure the width of a building. He identifies two points E and F at ground level on opposite sides of the building. He then walks to a point a distance away from the building: point G. He measures the distance from E to G as 284 feet, the distance from F to G as 322 feet, and the measure of ∠G as 63°. How wide is the building? Round your answer to the nearest foot, then determine whether it is reasonable given the context of the problem.

1. Draw and label a sketch of the situation.

2. Find EF using the Law of Cosines.

3. Determine if the answer is reasonable.

continued

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Example 2

Hideko is a pilot with a small two-seater plane. She took off from the airport and headed south toward her hometown. After flying for 70 miles, Hideko turned the plane 40° to the west to avoid a large storm. She remained on this course for 95 miles before turning back 65° to the east and continuing on to land near her hometown. How much farther did Hideko fly than she would have if she could have remained on her original course? Determine whether your answer is reasonable given the context of the problem.

Example 3

The Leaning Tower of Pisa, which is 55.8 meters tall, currently leans to the southeast. When the angle of elevation of the sun is 42° and the tower is leaning toward the sun, the tower’s shadow is 57.9 meters long. How many degrees does the tower lean? Round your answer to the nearest degree, then determine whether your answer is reasonable given the context of the problem.

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Problem-Based Task 3.2.3: Safe or Out?

A baseball diamond is a square with 90 feet in between each base. The pitcher’s mound is located on a direct line between home plate and second base and is 60.5 feet from home plate. During a pivotal play, the pitcher throws the ball to his teammate standing at first base at a speed of 132 feet per second in an attempt to get a runner out. The runner speeds back to the base in 0.6 second. Is the runner safe or out? Explain your reasoning. Use the equation for distance, d = rt, where d is distance, r is speed, and t is time. (Note: A runner is safe if he gets back to the base before the ball arrives. He is out if the ball is caught at the base before he returns.)

Is the runner safe or out?

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Answer the following questions using the Law of Sines or the Law of Cosines. Round answers to the nearest tenth.

1. Adam and Karla are standing by a silo. Karla heads due north of the silo and Adam heads 28 degrees west of north. When they stop, Adam is 72 yards away from the silo and 86 yards from Karla. How far is Karla from the silo?

2. A plane flew out of Las Vegas on a 1,515-mile flight to Chicago. When the plane was 723 miles away from Chicago, the plane experienced a technical problem and the pilot had to divert to the closest available airport, which was 42 miles away from the plane’s current location. When the plane landed, it was 701 miles away from Chicago. How many degrees did the pilot turn to divert to the closest airport?

3. A surveyor is trying to measure the width of a rock formation. He walks to a point far away from the formation and sets up his instruments. He measures the distance from the left side of the formation to his location as 89 feet and the distance from the right side of the formation to his location as 94 feet. He then measures the angle between these two lines of sight as 74°. How wide is the rock formation?

4. The sun is currently at a 67° angle of elevation. A tree is leaning away the sun at a 5° angle and casts an 18-foot shadow. How tall is the tree?

5. A ship sailed out of Baltimore on its way to Bermuda, which is 820 miles away. After 425 miles of smooth sailing, the captain decided to veer 35° to the left to avoid a storm. After sailing on that bearing for 167 miles, the captain turned the ship back on a direct line to Bermuda. How far must the ship travel to reach Bermuda?

Practice 3.2.3: Applying the Laws of Sines and Cosines

continued

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6. The sun is currently at an 82° angle of elevation. A 30-foot-tall telephone pole is leaning toward the sun at a 6° angle. How long is the pole’s shadow?

7. A surveyor identifies two points J and K on opposite sides of a valley. She then walks to a point a distance away from these points: point L. She measures the distance from J to L as 50 meters, the distance from K to L as 212 meters, and the measure of ∠L as 57°. What is the distance across the valley?

8. In a parallelogram, opposite angles and opposite sides have the same measure. The sides of a parallelogram are 6 cm and 4 cm long, and the angles are 135° and 45°. What are the lengths of the diagonals?

9. A flagpole that is 25 feet tall is leaning at a 2° angle away from the sun. When the flagpole’s shadow is 12 feet long, what is the angle of elevation of the sun?

10. Erika and Brad are both on the same street headed directly toward the Empire State Building, which is 1,454 feet tall. Erika, who is closer to the building, measures its angle of elevation from her location as 74°. Brad measures an angle of elevation. How far away is Erika from Brad?

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UNIT 3 • TRIGONOMETRY OF GENERAL TRIANGLES AND TRIGONOMETRIC FUNCTIONSLesson 3: Graphs of Trigonometric Functions

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Warm-Up 3.3.1

Spencer and Evan hang a cluster of bananas at the end of a spring, so that the bananas oscillate up and down once the spring is disturbed. They want to calculate how far the cluster will move from its resting position over time. This distance, known as the displacement from equilibrium, is given by the formula f (t) = cos 2t, in which f (t) is the displacement in feet and t is the time in seconds. Use this information to answer the questions that follow. Round all answers to the nearest hundredth. (Note: The formula uses radians.)

1. How far from equilibrium will the bananas be 1 second after the spring is disturbed?

2. How far from equilibrium will the bananas be 2 seconds after the spring is disturbed?

3. How far from equilibrium will the bananas be 3 seconds after the spring is disturbed?

Lesson 3.3.1: Periodic Phenomena and Amplitude, Frequency, and Midline

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Determine the period, frequency, midline, and amplitude of the graphed function.

2

1

–1

–2

–3

–4

–5

232

2 52

0 radians

y

1. Determine the period.

2. Determine the frequency.

3. Determine the midline.

4. Determine the amplitude.

continued

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Example 2

Determine the period, frequency, midline, and amplitude of the graphed function.

2

1

–1 232

20 radians

y

Example 3

Determine the period, frequency, midline, and amplitude of the function.

( ) 2 cos1

34f x x=

−

Example 4

Describe the difference between the function f(x) = 3 sin (x – 1) and the function g(x) = 3 sin [2(x – 1)].

88

What is the equation of the note that is one

octave higher than the note Jillian

played?



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Problem-Based Task 3.3.1: Perfect Pitch

In music, an octave is an interval between the first note and the eighth note on a musical scale. The first and eighth notes are the same, though one is higher than the other. Since sound travels in waves, a single note can be described by a sine curve, and the pitch of the note is determined by its frequency in hertz (Hz). If two notes are one octave apart, the higher note will have twice the frequency of the lower note. Jillian plays a note on the piano that can be described by the function f(x) = sin (880πt), where t is time in seconds. What is the equation of the note that is one octave higher than the note Jillian played?

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For problems 1–7, determine the period, frequency, midline, and amplitude of each trigonometric function.

1. 2

1

–1

–2

–3

–4

232

20 radians

y

2. 1

0

–1

–2

–3

–4

–5

–6

–7

232

2 52

3 72

4

radians

y

3. 3

2

1

–1 232

2

radians

y

0

4.

232

2

1

–1

0

y

radians

5. f (x) = cos (x + 5)

6. ( )1

2sin (2 ) 8f x x= −

7. ( ) 3 sin2

51 4f x x( )= −

+

For problems 8–10, the function for the periodic motion of a spring is given. Determine the frequency of the spring, or how many times it oscillates per second.

8. f (x) = 4 sin (6πt)

9. f (x) = 3 sin (πt)

10. ( ) sin3

2f x tπ=

Practice 3.3.1: Periodic Phenomena and Amplitude, Frequency, and Midline

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Warm-Up 3.3.2

Sound travels in waves, and a single note can be described by a sine curve. The larger the amplitude, the louder the note will be. Pitch refers to how high or low a note is. Since the pitch of a note is determined by its frequency, the greater the frequency, the higher the note will be.

Grant sat down at the piano and played a series of notes that can be described by the functions below. Use the given information about sound waves and the note functions to answer the questions.

• Note 1: f(x) = sin (880πt)

• Note 2: g(x) = 3 sin (784πt)

• Note 3: h(x) = 2.5 sin (1397πt)

• Note 4: j(x) = 2 sin (1568πt)

• Note 5: k(x) = 0.75 sin (392πt)

1. Which note was the loudest?

2. Which note was the quietest?

3. Which note was the highest in pitch?

4. Which note was the lowest in pitch?

Lesson 3.3.2: Using Trigonometric Functions to Model Periodic Phenomena

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Find the equation of a sine function with no horizontal shift whose frequency is 2. The function rises 3 units above its midline, which is y = –1.

1. Determine the value of a.

2. Determine the value of b.

3. Determine the value of c.

4. Determine the value of d.

5. Substitute a, b, c, and d into the general form of the sine function.

continued

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Example 2

Write an equation to describe the graphed function.

5

4

3

2

1

–1 232

2

radians

y

0

continued

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Example 3

The following graph shows historical average monthly temperatures for the town of Mayorsville starting in January 2000. Write an equation for the graphed function.

30

25

20

15

10

5

0

y

5 10 15 20 25

Tem

per

atu

re (

°C)

Month

x

Average MonthlyTemperature in Mayorsville

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Determine the cosine function that describes

the motion of the second spring.



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Problem-Based Task 3.3.2: Spring into Action

Ramzi observed the motion of a spring and plotted its displacement on the graph shown. He then observed a second spring that behaved in the same way except that its amplitude was twice as large as the first and its period was half as long. Determine the cosine function that describes the motion of the second spring.

4

3

2

1

0

–1

–2

–3

–4

y

2 4 6 8

Spring Displacementover Time in Seconds

Time (s)

Dis

pla

cem

ent

(in)

x

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For problems 1–7, write a trigonometric equation to describe each function. Note: Some problems may have more than one correct answer, but only one answer is needed.

1. A sine curve that is shifted 3

π units to the left has a midline at y = 2 and rises 4 units above the

midline. The length of one cycle of its curve is 2π.

2. A sine curve with no horizontal shift has a frequency of 2

π and rises 3 units above its midline,

which is at y = 1.

3. A cosine curve has two consecutive maximum points at 4

,4π

and 3

4,4

π

, and a minimum

point at 2

, 6π

−

.

4.

–1

–2

–3

232

20

y radians

5. 5

4

3

2

1

–1– 2 2

32

20

y

radians

6. 2

1

–1

–2

–3

–4

2 4 60

y

x

7. 3

2

1

–1 232

20

y

radians

Practice 3.3.2: Using Trigonometric Functions to Model Periodic Phenomena

continued

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For problems 8–10, write the trigonometric equation of the function that models each periodic phenomenon.

8. When Eve released a spring, it completed one oscillation (cycle) per second and traveled up to 4 inches from equilibrium (its midline). She knows the oscillation can be described by a cosine function with no horizontal or vertical displacement. Write an equation to describe this function.

9. Brandon played a note on his clarinet that had an amplitude of 2 and a frequency of 740 Hz. He knows this note can be described by a sine function with no horizontal or vertical displacement. Write an equation to describe this function.

10. Since January 2005, the average monthly temperatures of Swansburg have fluctuated in accordance with the function shown in the following graph. Write an equation to describe this function.

30

25

20

15

10

5

y

5 100 15 20

Average Monthly Temperature in Swansburg

Month

Tem

per

atu

re (°

C)

x

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UNIT 3 • TRIGONOMETRY OF GENERAL TRIANGLES AND TRIGONOMETRIC FUNCTIONSStation Activities Set 1: Trigonometric Functions

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CCSS IP Math III Teacher Resource © Walch Education

Station 1

Work with your group to complete the problems.

1. A unit circle has a radius of 1 unit. Explain why one complete rotation around a unit circle is a distance of 2π.

2. Complete the diagram that follows to show fractional rotations around a unit circle in radians. Remember, every rotation starts at (1, 0) and goes in a counterclockwise direction.

y

x1 unit

43

6

3. Angles can be measured in radians as well as in degrees. A complete rotation of a circle is 360°. A complete rotation also equals 2π radians. On the diagram for problem 2, write the degree measure next to each radian measure.

continued

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4. Repeat problems 2 and 3 for angles measured in a clockwise direction. Use the figure that follows. Label each location on the unit circle with two measures. The first two locations have been labeled for you.

y

x1 unit

, –30˚

, –45˚4

6

5. The table that follows shows the special angles in Quadrant I. Complete the table, including both degree and radian angle measures.

Exact Trigonometric Values for Special Angles in Quadrant I

Degrees 0° 30° 45° 60° 90°

Radians

Sine

Cosine

6. Explain how to change a degree measure to a radian measure.

7. Explain how to change a radian measure to a degree measure.

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Station 2

Work with your group and refer to the following figures to complete the problems. In each figure, a lightly shaded reference angle in the stated quadrant is shown for a given (darkly shaded) angle.

y

x

Quadrant IIy

x

Quadrant IIIy

x

Quadrant IV

1. The degree measure of an angle in the third quadrant is given. Explain how to find the reference angle.

2. An angle in the fourth quadrant is measured in radians. Explain how to find the reference angle.

3. A student subtracted an angle measure from 180° to find the reference angle. In which quadrant was this angle?

4. An angle that has completed less than a full negative rotation with a negative measure lies in Quadrant IV. Explain how to find the reference angle.

5. Choose three angles in Quadrant II. Write the measures in degrees. Find the reference angle for each one.

continued

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6. Choose three angles in Quadrant III. Write the measures in degrees. Find the reference angle for each one.

7. Choose three angles in Quadrant IV. Write the measures in degrees. Find the reference angle for each one.

8. Choose three angles in Quadrant II. Write the measures in radians. Find the reference angle for each one.

9. Choose three angles in Quadrant III. Write the measures in radians. Find the reference angle for each one.

10. Choose three angles in Quadrant IV. Write the measures in radians. Find the reference angle for each one.

11. Explain why 0° can be considered the reference angle for 180°. Which other pair of degree measures have this same relationship?

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Station 3


1. Study the triangles that follow. The hypotenuse of each triangle is 1 unit long. Use the Pythagorean Theorem to find the exact values of all side lengths. (Hint: An exact value may have a square root sign in it.)

11 1

1

45˚ 30˚ 60˚

2. How do you use the side lengths of a right triangle to find values for the sine and cosine functions?

continued

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3. Complete the following tables to show the exact and approximate values for the sine and cosine of the given angles.

Exact Trigonometric Values for Special Angles

Angle 0° 30° 45° 60° 90°

Sine 0

Cosine 1 32

Approximate Trigonometric Values for Special Angles

Angle 0° 30° 45° 60° 90°

Sine 0.71

Cosine 0.50

4. What patterns are found in the tables? How can these patterns help you memorize these key values?

continued

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5. The tables that follow show the multiples of 30°, 45°, and 60° angles. Complete the tables with exact trigonometric values for the sine and cosine.

Exact Trigonometric Values for Multiples of 30° Angles

Angle 30° 90° 150° 210° 330°

Sine12

− 12

Cosine − 32

32


Angle 45° 90° 135° 225° 315°

Sine

Cosine


Angle 60° 120° 240° 300°

Sine

Cosine

continued

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6. The tables that follow show the multiples of –30°, –45°, and –60° angles. Complete the tables with exact trigonometric values for the sine and cosine.

Exact Trigonometric Values for Multiples of –30° Angles

Angle –330° –210° –150° –30°

Sine

Cosine


Angle –315° –225° –135° –45°

Sine

Cosine


Angle –300° –240° –120° –60°

Sine

Cosine

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Station 4


1. Explain how the sine and cosine functions can be used to write ordered pairs for points on the unit circle. Use the diagram that follows in your explanation.

x

P(cos θ, sin θ)

(1, 0)0

θx

y

2. The bullets on the following unit circle show some multiples of 45° angles. The angles are marked in radians. Use exact values for the sine and cosine functions to write the coordinates of each point.

y

x1 unit

3

4

5

4

4

7

4

continued

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3. The following unit circle shows some multiples of 30° and 60° angles, measured in radians. Use exact values for the sine and cosine functions to write the coordinates of each point.

2

3 3

4

3

5

3

5

6

7

6

11

6

6

y

x1 unit

4. The chart that follows shows the special angles in the first quadrant of the unit circle, with angle measures in radians. Write ordered pairs to show the location of the intersection points on the unit circle for rays that create these angles.

Angle 0π6

π4

π3

π2

Location on the unit circle

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UNIT 3 • TRIGONOMETRY OF GENERAL TRIANGLES AND TRIGONOMETRIC FUNCTIONSStation Activities Set 2: The Laws of Sines and Cosines

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Station 1

For problems 1 and 2, use the trigonometric ratios of sine, cosine, and tangent to find the missing side length for each triangle shown. Round answers to the nearest tenth.

1. Determine the length of side a.

A

B

C

a38 m

48˚

2. Determine the length of side a.

A B

C

a

83 in20˚

continued

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For problems 3–5, use the information given in each scenario to complete the problems.

3. Ana and Liam each found a different length for side c in the triangle shown in the following diagram. Compare Ana’s work with Liam’s work to determine whose answer is correct. Justify your response.

12 ftc

36˚

Ana’s work Liam’s work

cos 3612

°=c

sin 3612

°=c

c cos 36° = 12 c sin 36° = 12

12

cos36=

°c

12

sin36=

°c

c ≈ 14.83 c ≈ 20.42

4. A 40-foot ladder leans against a tree. The ladder forms a 75° angle with the ground. How far from the tree is the base of the ladder?

5. ABC is a right triangle. ∠C is a right angle, 50∠ =m A , and AB = 40 cm.

a. Sketch and label the triangle.

b. Determine the lengths of AC and BC .

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Station 2

Use trigonometric ratios and the provided materials, as needed, to complete the problems.

1. Derive the formula for the area of a triangle using trigonometry and the figure shown in the diagram. As a starting point, use one of the trigonometric ratios (sine, cosine, or tangent) to identify the height, h.

a

bh

c

A

BC

For problems 2–4, use the paper and straws or licorice sticks provided to construct each triangle described. Then, find the area of the triangle to the nearest tenth of a unit.

2. Construct a triangle with side lengths of 10 cm and 12 cm; the included angle measures 70°.

3. Construct a triangle with side lengths of 4 in and 5 in; the included angle measures 50°.

4. has adjacent side lengths of 40 cm and 50 cm; the included angle measures 100°.

continued

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For problems 5 and 6, sketch each triangle described, then use the formula found in problem 1 to determine the area of the triangle. As you sketch and construct the triangles using the straws and licorice sticks, remember to label the side opposite ∠A with a, the side opposite ∠B with b, and the side opposite ∠C with c.

5. has two sides that each measure 10 cm and a non-included angle of 60°.

6. has adjacent sides with lengths of 10 cm and 16 cm and an included angle of 40°.

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Station 3

At this station, you will find paper and straws or licorice sticks. Use these materials, as needed, as well as the Law of Sines and the Law of Cosines to complete each problem.

For problems 1–4, state whether the Law of Sines or the Law of Cosines should be used to solve each problem.

1. Given the lengths of three sides of a triangle, find the measure of an angle.

2. Given the lengths of two sides of a triangle and the measure of an angle opposite one of those sides, find the measures of the two unknown angles.

3. Given the lengths of two sides of a triangle and the measure of an included angle, find the length of the side opposite the known angle measure.

4. Given the measure of two angles of a triangle and the length of one side, find the lengths of the missing sides.

For problems 5 and 6, use the paper and straws or licorice sticks provided to construct the triangle described, then use the construction to answer the questions.

5. Construct a triangle with side lengths of 10 cm and 12 cm; the included angle measures 70°.

a. Use trigonometric ratios to determine the length of the third side of the triangle (the side opposite the 70° angle).

b. Verify the length of the third side using the Law of Sines or the Law of Cosines. What do you notice about your answer?

continued

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6. Construct a triangle with side lengths of 4 in and 5 in; the included angle measures 50°.

a. Use trigonometric ratios to determine the length of the third side of the triangle (the side opposite the 50° angle).

b. Verify the length of the third side using the Law of Sines or the Law of Cosines. What do you notice about your answer?

For problems 7 and 8, create a sketch of each triangle described, then use your sketch to answer the question.

7. For ABC , AC = 12 cm, CB = 10 cm, and 50∠ =m C . What is the area of ABC ?

8. For ABC , AC = 14 cm, CB = 22 cm, and the area is 87 cm2. What is the measure of ∠C ?

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Station 4

Sketch each triangle described and then apply the Law of Sines or the Law of Cosines to complete each problem. Unless otherwise indicated, round answers to the nearest tenth.

1. Clevon forms a triangle with three straws measuring 20, 25, and 18 inches. To the nearest whole degree, find the measure of the angle enclosed by the 20- and 25-inch straws.

2. Olivia flies a kite on a 1,850-foot string at an angle of elevation of 80° with the ground. An observer notes that the angle formed between the kite and Olivia is 100°. How far is the observer from the kite?

3. Two stabilizing wires extend from the top of a pole to the ground, forming right triangles on either side of the pole. One wire forms an angle of 38° with the ground and the other wire forms an angle of 67° with the ground. The distance between where each of the wires attaches to the ground is 30 meters. How tall is the pole?

continued

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4. A plane took off from an airport and flew in a straight path for 2 hours 30 minutes. The pilot then headed 9° to the west and flew 2 hours in the new direction before landing the plane. If the pilot maintained an average speed of 350 mph, how far away from the starting point did he land?

5. Two planes took off in opposite directions from one airport at the same time. One plane flew at an average speed of 250 mph and the other flew at an average speed of 450 mph. After 2 hours, the planes were 1,000 miles apart. What is the measure of the angle between their flight paths at that time?

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