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Faculty of Science

Unit 5: Introduction to

Trigonometry

MATH 1001 Pre-Calculus Mathematics

reserved.

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TRU seeks to ensure that any course content that is owned by others has been

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Select questions associated with the following course textbook are used with

permission from Cengage Learning:

Swokowski, E. W., & Cole, J. A. (2012). Precalculus: Functions and graphs

(12th ed.). Belmont, CA: Brooks/Cole, Cengage Learning

Course Revision Team (2019)

Course Reviser: Saeed Rahmati, PhD

Course Editor: Courtney Charlton, MA

Associate Dean, Science: Dennis Acreman, PhD

Course Development Team

Course Writer: Alan Cooper, PhD

Consultant: Bernadette Harris, PhD

Course Reviser (2014): Bernadette Harris, PhD

Course Revisions/Writing (2008): Fae Debeck, MS, and Adriana Stefan, MMath

Instructional Designer: Ted Keating, MEd

Course Reference: MATH 1001_SW3

Thompson Rivers University

805 TRU Way

Kamloops, BC

V2C 0C8

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Table of Contents Unit 5: Introduction to Trigonometry

Introduction .................................................................................................................. U5‐1

Learning Outcomes ...................................................................................................... U5‐1

5.1: Angles ..................................................................................................................... U5‐2

Introduction .............................................................................................................. U5‐2

Section Learning Outcomes .................................................................................... U5‐2

Study Plan ................................................................................................................ U5‐2

Sample Questions from Section 5.1 ....................................................................... U5‐3

Further Practice Section 5.1 ..................................................................................... U5‐4

5.2: Trigonometric Functions of Angles .................................................................. U5‐5

Introduction .............................................................................................................. U5‐5

Section Learning Outcomes .................................................................................... U5‐5

Study Plan ................................................................................................................. U5‐5

Sample Questions from Section 5.2 ....................................................................... U5‐7

Further Practice Section 5.2 ..................................................................................... U5‐9

5.3: Trigonometric Functions of Real Numbers .................................................. U5‐10

Introduction ............................................................................................................ U5‐10

Section Learning Outcomes .................................................................................. U5‐10

Study Plan ............................................................................................................... U5‐10

Sample Questions from Section 5.3 ..................................................................... U5‐12

Further Practice Section 5.3 ................................................................................... U5‐14

5.4: Values of the Trigonometric Functions ......................................................... U5‐15

Introduction ............................................................................................................ U5‐15

Section Learning Outcomes .................................................................................. U5‐15

Study Plan ............................................................................................................... U5‐15

Sample Questions from Section 5.4 ..................................................................... U5‐16

Further Practice Section 5.4 ................................................................................... U5‐17

5.5: Trigonometric Graphs ....................................................................................... U5‐18

Introduction ............................................................................................................ U5‐18

Section Learning Outcomes .................................................................................. U5‐18

Study Plan ............................................................................................................... U5‐18

Sample Questions from Section 5.5 ..................................................................... U5‐19

Further Practice Section 5.5 ................................................................................... U5‐21

5.6: Additional Trigonometric Graphs .................................................................. U5‐22

Introduction ............................................................................................................ U5‐22

Section Learning Outcomes .................................................................................. U5‐22

Study Plan ............................................................................................................... U5‐22

Sample Questions from Section 5.6 ..................................................................... U5‐23

Further Practice Section 5.6 ................................................................................... U5‐25

5.7: Applied Problems .............................................................................................. U5‐26

Introduction ............................................................................................................ U5‐26

Section Learning Outcomes .................................................................................. U5‐26

Study Plan ............................................................................................................... U5‐26

Sample Questions from Section 5.7 ..................................................................... U5‐27

Further Practice Section 5.7 ................................................................................... U5‐28

MATH 1001: Pre-Calculus Mathematics U5-1

TRU Open Learning

Unit 5: Introduction to Trigonometry This unit corresponds to Chapter 5 of the textbook. In Unit 5 you will be covering all

of the sections from Chapter 5 in your textbook.

 5.1: Angles

 5.2: Trigonometric Functions of Angles

 5.3: Trigonometric Functions of Real Numbers

 5.4: Values of the Trigonometric Functions

 5.5: Trigonometric Graphs

 5.6: Additional Trigonometric Graphs

 5.7: Applied Problems

Introduction The word trigonometry translates from Greek literally as, “measurement of

triangles,” but although triangles occur throughout the subject, it is also the basic

tool for understanding relationships between lengths and angles in any context, and

so is an invaluable tool in many geometric and physical problems.

Learning Outcomes In this unit you will study the trigonometric functions as functions of both angles

and real numbers. Specific learning outcomes are listed at the beginning of each

section of the unit.

U5-2 Unit 5: Introduction to Trigonometry

TRU Open Learning

5.1: Angles

Introduction

This section reviews various ways of measuring angles. We introduce and

emphasize a unit of measurement that you may not have seen before (the radian)

but which is convenient in situations where we want simple relations between

lengths and angles.

Section Learning Outcomes

After completing this section you should be able to:

 Draw angles in standard position.

 Determine co‐terminal angles for a given angle.

 Convert degree and radian measure from one to another.

 Solve geometric and applied problems using arc length and sector formulas.

Study Plan

1. Read section 5.1 of the textbook.

2. Read the following “Study Notes and Sample Questions” section to prepare for

the practice exercises.

3. Follow the instructions regarding “Further Practice.”

Study Notes and Sample Questions

Not all of the material presented in each section of the textbook is required for this

course and some topics are more important than others. To master the required

material in this section, review the following topics and examples and the

supplemental “Sample Questions from Section 5.1”.

MATH 1001: Pre-Calculus Mathematics U5-3

TRU Open Learning

Section 5.1: Angles

Topic Page, Example Numbers

Angles Page 324

Finding Co‐terminal Angles Page 325—Example 1

Special Types of Angles Page 325—Table

Finding Complementary Angles Page 326—Example 2

Definition of Radian Measure Page 326

Relationships Between Degrees and

Radians

Page 327

Changing Angular Measures Page 327—Table

Radians/Degrees Page 327—Table

Formula for the Length of a Circular

Arc

Page 329

Formula for the Area of a Circular Arc Page 329

Using the Circular Arc and Sector

Formulas

Page 330

Sample Questions from Section 5.1

Page 332—q. 36

(a) Find the length of the arc that subtends the given central angle 2.2  on a circle of diameter 120 cmd  .

(b) Find the area of the sector determined by 2.2  ; 120 cmd 

U5-4 Unit 5: Introduction to Trigonometry

TRU Open Learning

Solution:

(a)     1 120 2.2 60 2.2 132 cm 2

s r   

(b)    22 21 1 60 2.2 3, 960 cm 2 2

A r   

Page 333—q. 50

Tire Revolutions

A typical tire for a compact car is 22 inches in diameter. If the car is traveling at a

speed of 60 mi/hr, find the number of revolutions the tire makes per minute.

Solution:

60 60 1 5280 12 63, 360

miles miles hour feet inches inches

hour hour minutes mile foot minute     

In one revolution a point of the circumference of the tire moves:

1 2 2 22 22

2 r     

  inches.

Thus, the number of revolutions per minute is:

63, 360 1 31, 680 916.73

22 11

inches revolution revolutions rpm

minute inches minutes     .

Further Practice Section 5.1

Check your understanding and improve your speed by working through some of

the exercises on pages 331–332 of the textbook. Do enough of the odd‐numbered

questions of each type to convince yourself that you can get the right answers. Note

that the answers are at the back of the textbook and complete worked‐out solutions

are in the Student Solutions Manual––but try to avoid looking at answers or solutions

until you have made your own best effort.

As a minimum you should do questions 1, 3, 5, 9, 11, 13, 31, 33, and 35 from Section

5.1. When done, compare your solutions with those in the Student Solutions Manual.

MATH 1001: Pre-Calculus Mathematics U5-5

TRU Open Learning

5.2: Trigonometric Functions of Angles

Introduction

In this section we will introduce the trigonometric functions as ratios of sides in a

right triangle.

Section Learning Outcomes

After completing this section you should be able to:

 Find the values of all six trigonometric ratios for any angle in a right‐angled triangle when at least two of the side lengths are given.

 State the trigonometric ratios as exact values for the special angles— 60 , 45 , and 30 —and use them to solve applied problems.

 Use a calculator to determine approximate trigonometric ratios for any given angle.

 Use the reciprocal and Pythagorean identities to verify other identities.

 Determine the trigonometric function values for any angle in standard position including the quadrantal angles.

 Determine the quadrant containing the terminal side of a given angle.

Study Plan

1. Read section 5.2 of the textbook.

2. Read the following “Study Notes and Sample Questions” section to prepare for

the practice exercises.

3. Follow the instructions regarding “Further Practice.”

Study Notes and Sample Questions

Not all of the material presented in each section of the textbook is required for this

course and some topics are more important than others. To master the required

material in this section, review the following topics and examples and the

supplemental “Sample Questions from Section 5.2”.

U5-6 Unit 5: Introduction to Trigonometry

TRU Open Learning

Section 5.2: Trigonometric Functions of Angles

Topic Page, Example Number

Definition of the Trigonometric Functions

of an Acute Angle of a Right Triangle

Page 334

Reciprocal Identities Page 335

Finding Trigonometric Function Values Page 335

Finding Trigonometric Function Values

of 60 , 30 , and 45 Page 336

Special Values of Trigonometric

Functions

Page 336—Table

The Fundamental Identities Page 338—Table

Using Fundamental Identities Page 339—Example 4

Showing That an Equation is an Identity Page 340—Example 5

Verifying an Identity Page 341—Example 6

Definition of the Trigonometric Functions

of Any Angle

Page 342

Finding Trigonometric Function Values of

an Angle in Standard Position

Page 343—Examples 7, 8

Finding Trigonometric Function Values of

a Quadrantal Angle

Page 344—Example 9

Finding the Quadrant Containing an

Angle

Page 345—Example 10

Finding Values of Trigonometric

Functions from Prescribed Conditions

Page 345—Example 11

Using Fundamental Identities Page 345—Example 12

MATH 1001: Pre-Calculus Mathematics U5-7

TRU Open Learning

Sample Questions from Section 5.2

Page 346—q. 18

Find the exact values of the trigonometric functions for the acute angle  given that 8

cos 17

  .

Solution:

8 cos

adj

hyp   

 22 28 17 289 64 15opp opp     

8 15 17adj opp hyp   

So 15 17 17 15 8

sin csc sec tan cot 17 15 8 8 15

        

Page 347—q. 36

Use the Pythagorean identities to write the expression as an integer

a) 2 2csc 3 cot 3x x 

b) 2 23csc 3cotx x 

Solution:

a) Since 2 2 2 21 cot 3 csc 3 csc 3 cot 3 1x x x x    

b)    2 2 2 23csc 3cot 3 csc cot 3 1 3x x x x    

Page 348—q. 43

Simplify the expression: 2 tan

2 csc sec

   

 

Solution:

sin 2 cos sin 2

2 tan 2 cos sin sin coscos cos sin 1 1 2 cos sin2 csc sec cos 2 cos sin2

sin cos sin cos

         

         

  

       

U5-8 Unit 5: Introduction to Trigonometry

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Page 348—q. 60

Verify the identity by transforming the left‐hand side into the right hand‐side:

  2tan cot tan sec    

Solution:

    2 2

2 2 2 2

2 2 2

tan cot tan tan cot tan tan 1

sin 1 sin cos 1 sec

cos 1 cos cos

      

   

  

     

     

Page 348—q. 74

Verify the identity:      log tan log sin log cos   

Solution:

Transform the left‐hand side into the right‐hand side:

     sinlog tan log log sin log cos cos

   

       

Page 349—q. 93

Use fundamental identities to find the values of the six trigonometric functions for

the given conditions:

1 cos and sin 0

3    

Solution:

1 cos and sin 0

3       is in Quadrant III

2 2 1 1 8 2 2sin 1 cos 1 1

3 9 3 3                  

  2 2

3  

2 2 sin 3tan 2 2

1cos 3

 



   



1 1 2 cot

tan 42 2 

   

MATH 1001: Pre-Calculus Mathematics U5-9

TRU Open Learning

1 sec 3

cos 

   

1 3 3 2 csc

sin 42 2 

     

3 2

4  

Further Practice Section 5.2

Check your understanding and improve your speed by working through some of

the exercises on pages 346–349 of the textbook. Do enough of the odd‐numbered

questions of each type to convince yourself that you can get the right answers. Note

that the answers are at the back of the textbook and complete worked‐out solutions

are in the Student Solutions Manual––but try to avoid looking at answers or solutions

until you have made your own best effort.

As a minimum you should do questions 3, 5, 9, 11, 13, 15, 17, 19, 21, 35, 39, 49, 51, 53,

57, 75, 77, 79, 81, 83, 87, and 89 from Section 5.2. When done, compare your solutions

with those in the Student Solutions Manual.

U5-10 Unit 5: Introduction to Trigonometry

TRU Open Learning

5.3: Trigonometric Functions of Real Numbers

Introduction

In this section, you will study the trigonometric functions as functions of real

numbers rather than of angles. Trigonometric functions using radians measure are

more convenient for many purposes–in particular for calculus and for modeling

applied problems involving periodic phenomena.

Section Learning Outcomes

After completing this section you should be able to:

 Use the unit circle to evaluate the trigonometric function values of any real number.

 Determine the exact values for the trigonometric functions for the special numbers of 0, 6, 4 , 3, 2 and      .

 Use the properties of even and odd functions to determine function values.

Study Plan

1. Read section 5.3 of the textbook

2. Read the following “Study Notes and Sample Questions” section to prepare for

the practice exercises.

3. Follow the instructions below regarding “Further Practice.”

Study Notes and Sample Questions

Not all of the material presented in each section of the textbook is required for this

course and some topics are more important than others. To master the required

material in this section, review the following topics and examples and the

supplemental “Sample Questions from Section 5.3.”

MATH 1001: Pre-Calculus Mathematics U5-11

TRU Open Learning

Section 5.3: Trigonometric Functions of Real Numbers

Topic Page, Example Numbers

Definition of the Trigonometric

Functions of Real Numbers

Pages 349–350

Definition of the Trigonometric

Functions in Terms of a Unit Circle

Page 351

Finding Values of the Trigonometric

Functions

Page 351—Example 1

Finding a Point on U Relative to a

Given Point

Page 351—Example 2

Finding Special Values of the

Trigonometric Functions

Pages 352–353—Example 3

Theorem on Repeated Function Values

for sin and cos

Page 354

Sketching the Graphs of the Sine and

Cosine Functions

Pages 354, 355

Formulas for Negatives Page 355

Use of Formulas for Negatives Page 356—Illustration

Using Formulas for Negatives to Verify

an Identity

Page 356—Example 4

Theorem on Even and Odd

Trigonometric Functions

Page 357

Graph of tany x Page 357

Page 358—Figure 16

Solving Equations Involving a

Trigonometric Function

Page 361—Example 6 (a)

U5-12 Unit 5: Introduction to Trigonometry

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Sample Questions from Section 5.3

Page 363—q. 6

Let 8 15

( ) , 17 17

P t      

be the point on the unit circle U that corresponds to the real

number t. Find the following points.

(a) ( )P t  (b) ( )P t 

Solution:

(a) 8 15

( ) , 17 17

P t        

(b) 8 15

( ) ( ) , 17 17

P t P t          

( ) , 17 17

P t        

(d)   8 15( ) ( ) , 17 17

P t P t            

Page 363—q. 10

Let P be the point on the unit circle U that corresponds to the real number t. Find the

coordinates of P and the exact values of the trigonometric functions of t, whenever

possible.

(a)  (b) 6 Solution:

(a)    ( ) 1, 0 cos , sint P t t t      0

sin 0 cos 1 tan 0 1

t t t      

1 1 1 csc undefined sec 1 cot undefined

0 1 0 t t t

       



(b)    6 ( ) 1, 0 cos , sint P t t t    0

sin 0 cos 1 tan 0 1

t t t    

1 1 1 csc undefined sec 1 cot undefined

0 1 0 t t t     

MATH 1001: Pre-Calculus Mathematics U5-13

TRU Open Learning

Page 364—q. 18

Use a formula for negatives to find the exact values of:

(a) 3

sin 2

    

(b)  cos 225  (c)  tan 

Solution:

(a)  3 3sin sin 1 1 2 2

           

(b)   2cos 225 cos 225 2

     

Page 364—q. 26

Verify the identity by transforming the left‐hand side into the right‐hand side:

     cot cos sin cscx x x x     

Solution:

     

2 2

cot cos sin cot cos sin

cos cos sin

sin

cos sin

sin 1

csc sin

x x x x x x

x x x

x x

      

  

  

   

Page 364—q. 40

Refer to the graph of siny x or cosy x to find the exact values of x in the interval [0, 4 ] , that satisfy the equation: sin 1x 

Solution:

2 x

  and

5 2

2 2 x

    

U5-14 Unit 5: Introduction to Trigonometry

TRU Open Learning

Further Practice Section 5.3

Check your understanding and improve your speed by working through some of

the exercises on pages 363–365 of the textbook. Do enough of the odd‐numbered

questions of each type to convince yourself that you can get the right answers. Note

that the answers are at the back of the textbook and complete worked‐out solutions

are in the Student Solutions Manual––but try to avoid looking at answers or solutions

until you have made your own best effort.

As a minimum you should do questions 5, 7, 9, 11, 17, 21, 23, 39, 41, 43, 51, 53, and 55

from Section 5.3. When done, compare your solutions with those in the Student

Solutions Manual.

MATH 1001: Pre-Calculus Mathematics U5-15

TRU Open Learning

5.4: Values of the Trigonometric Functions

Introduction

In Sections 5.2 and 5.3 we calculated the values of the trigonometric functions using

triangles or the unit circle. In practice, we most often use a calculator to approximate

these values. In this section you will learn how to use a reference angle to find all

angles corresponding to a given function value.

Section Learning Outcomes

After completing this section you should be able to:

 Determine the reference angle for any given angle in either degree or radian measure.

 Use a calculator to determine an angle given particular trigonometric function values.

Study Plan

1. Read section 5.4 of the textbook.

2. Read the following “Study Notes and Sample Questions” section to prepare for

the practice exercises.

3. Follow the instructions regarding “Further Practice.”

Study Notes and Sample Questions

Not all of the material presented in each section of the textbook is required for this

course and some topics are more important than others. To master the required

material in this section, review the following topics and examples and the

supplemental “Sample Questions from Section 5.4”.

U5-16 Unit 5: Introduction to Trigonometry

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Section 5.4: Values of the Trigonometric Functions

Topic Page, Example Number

Definition of Reference Angle Page 366

Page 366—Figure 1

Finding the Reference Angle Page 367—Example 1

Theorem of the Reference Angle Page 368

Using Reference Angles Page 368—Example 2

Problem Page 369

Finding Acute Angle and

Solutions of Equations with a

Calculator

Page 370—Illustration

Page 370—Table

Finding Angles with a Calculator Page 370—Illustration

Approximating an Angle with a

Calculator

Page 371—Examples 3, 4

Sample Questions from Section 5.4

Page 372—q. 4

Find the reference angle R if  has the given measure:

(a) 7

 (b)



  (d)



Solution:

(a) 7

 is in Q IV

7 2

4 4 R

      

(b) 2

 is in Quadrant II

3 3 R

      

  is co‐terminal with

 and

 is in Q III

4 4 R

      

MATH 1001: Pre-Calculus Mathematics U5-17

TRU Open Learning

(d) 23

  is co‐terminal with

 and

 is in Q I

6 R

  

Page 372—q. 10

Find the exact value:

(a) 5

cos 4

     

(b) 11

cos 6

    

Solution:

(a) 5 2

cos cos 4 4 2

              

(b) 11 3

cos cos 6 6 2

             

Page 372—q. 26

Given that sin 0.6612  , approximate the acute angle  to the nearest 0.01 .

Solution:

 1sin 0.6612 sin 0.6612 41.39       .

Further Practice Section 5.4

Check your understanding and improve your speed by working through some of

the exercises on page 372 of the textbook. Do enough of the odd‐numbered questions

of each type to convince yourself that you can get the right answers. Note that the

answers are at the back of the textbook and complete worked‐out solutions are in the

Student Solutions Manual––but try to avoid looking at answers or solutions until you

have made your own best effort.

As a minimum you should do questions 1, 3, 5, 7, 9, 11, 27, and 29 from Section 5.4.

When done, compare your solutions with those in the Student Solutions Manual.

U5-18 Unit 5: Introduction to Trigonometry

TRU Open Learning

5.5: Trigonometric Graphs

Introduction

Because of their periodicity, trigonometric functions are useful for modeling all

kinds of periodic or cyclical phenomena. In order to match any particular oscillation,

it may be necessary to shift or scale the basic sine or cosine graphs we sketched in

the last section. In this section you will expand upon the basic techniques of shifting

and scaling learned in Unit 2.

Section Learning Outcomes

After completing this section you should be able to:

 Sketch the graphs of the basic sine and cosine functions.

 Use amplitude, period, and shifts to sketch more complex trigonometric graphs involving sine and cosine functions.

 Use amplitude, period, and shifts to model and solve applied problems.

Study Plan

1. Read section 5.5 of the textbook.

2. Read the following “Study Notes and Sample Questions” section to prepare for

the practice exercises.

3. Follow the instructions regarding “Further Practice.”

Study Notes and Sample Questions

Not all of the material presented in each section of the textbook is required for this

course and some topics are more important than others. To master the required

material in this section, review the following topics and examples and the

supplemental “Sample Questions from Section 5.5”.

MATH 1001: Pre-Calculus Mathematics U5-19

TRU Open Learning

Section 5.5: Trigonometric Graphs

Topic Page, Example Number

Graphs of:

1 sin ; 2 sin ; sin

2 y x y x y x  

Page 374—Figures 1, 2

Sketching the Graph of an Equation

Involving sin x Page 374—Example 1

Sketching the Graph of an Equation

Involving cos x Page 375—Example 2

Theorem on Amplitudes and

Periods

Page 375

Finding an Amplitude and a Period Pages 375–376—Examples 3, 4

Finding an Amplitude and a Period Pages 376–377—Examples 5, 6

Vertically Shifting a Trigonometric

Graph

Page 377—Example 7

Theorem of Amplitudes, Periods

and Phase Shifts

Page 377

Finding an Amplitude, a Period and

a Phase Shift

Page 378—Examples 8, 9

Finding an Equation for a Sine Wave Page 379—Example 10

Sample Questions from Section 5.5

Page 383—q. 8

Find the amplitude, the period and the phase shift and sketch the graph of the

equation:

2 sin 3

y x    

 

U5-20 Unit 5: Introduction to Trigonometry

TRU Open Learning

Solution:

Amplitude 2 2  Phase Shift 3

 

Period 2 One cycle 7

, 3 3

     

Page 383—q. 28

Find the amplitude, the period and the phase shift and sketch the graph of the

equation:

4 cos 2 3

y x     

 

Solution:

4 cos 2 4 cos 2 4 cos 2 3 6 6

y x x x                                      

Amplitude 4 4   Phase Shift 6

  

Period 2

   One cycle =

5 ,

6 6

    

MATH 1001: Pre-Calculus Mathematics U5-21

TRU Open Learning

Page 384—q. 46

Intensity of Daylight

On a certain spring day with 12 hours of daylight, the light intensity I takes on its

largest value of 510 calories/cm2 at midday. If 0t  corresponds to sunrise, find a formula sinI a bt that fits the information:

Solution:

12 24 hours b

    

The light intensity would be 0 at 0t  (sunrise) and 12t  (sunset).

The light intensity is at its maximum value of 510 at 6t  .

510 sin 12

I t    

 

Further Practice Section 5.5

Check your understanding and improve your speed by working through some of

the exercises on pages 382–383 of the textbook. Do enough of the odd‐numbered

questions of each type to convince yourself that you can get the right answers. Note

that the answers are at the back of the textbook and complete worked‐out solutions

are in the Student Solutions Manual––but try to avoid looking at answers or solutions

until you have made your own best effort.

As a minimum you should do questions 1 (a), (b), (g), (h), 3 (a), (b), (g), (h), 5, 7, 9,

and 41 from Section 5.5. When done, compare your solutions with those in the

Student Solutions Manual.

U5-22 Unit 5: Introduction to Trigonometry

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5.6: Additional Trigonometric Graphs

Introduction

In the previous section, transformations of the graphs of the sine and the cosine

functions were discussed. These same ideas can be applied to the graphs of the other

four trigonometric functions. In this section you will learn how to apply

transformations to the graph of the tangent function. You will also learn how to

sketch the graph of the absolute value of a trigonometric function.

Section Learning Outcomes

After completing this section you will be able to:

 Sketch the graph of the basic tangent function.

 Use transformations to sketch more complex trigonometric graphs involving the tangent function.

 Sketch the graph of the absolute value of a trigonometric function.

Study Plan

1. Read section 5.6 of the textbook.

2. Read the following “Study Notes and Sample Questions” section to prepare for

the practice exercises.

3. Follow the instructions regarding “Further Practice.”

Study Notes and Sample Questions

Not all of the material presented in each section of the textbook is required for this

course and some topics are more important than others. To master the required

material in this section, review the following topics and examples and the

supplemental “Sample Questions from Section 5.6”.

MATH 1001: Pre-Calculus Mathematics U5-23

TRU Open Learning

Section 5.6: Additional Trigonometric Graphs

Topic Page Example, Number

Sketching the Graph of an

Equation Involving tan x Page 386—Example 1

Theorem on the Graph of

 tany a bx c  Page 387

Sketching the Graph of an

Equation of the Form

 tany a bx c 

Page 387—Example 2

Sketching the Graph of an

Equation Involving an Absolute

Value

Page 390—Example 6

Sample Questions from Section 5.6

Page 391—q. 16

Find the period and sketch the graph of the equation. Show the asymptotes:

1 tan 2

3 4 y x

     

Solution:

1 1 tan 2 tan 2

3 4 3 8 y x x

                

One cycle of the graph of tany x is in , 2 2

     

so for this function we have

2 2 4 2

x   

    3 3

2 4 4 8 8

x x    

      

So one cycle is in 3

, 8 8

     

and the period is 2

 .

U5-24 Unit 5: Introduction to Trigonometry

TRU Open Learning

The vertical asymptotes are all vertical lines of equation: 8 2

x n  

   where

0, 1, 2,n    

Page 392—q. 62

Use the graph of a trigonometric function to aid in sketching the graph of the

equation without plotting points.

cos 3y x 

Solution:

Start with the graph

of cosy x

Graph cosy x

MATH 1001: Pre-Calculus Mathematics U5-25

TRU Open Learning

Shift it down 3 units

to get the graph of

cos 3y x 

Further Practice Section 5.6

Check your understanding and improve your speed by working through some of

the exercises on pages 391–392 of the textbook. Do enough of the odd‐numbered

questions of each type to convince yourself that you can get the right answers. Note

that the answers are at the back of the textbook and complete worked‐out solutions

are in the Student Solutions Manual––but try to avoid looking at answers or solutions

until you have made your own best effort.

As a minimum you should do questions 1, 9, 11, 13, 15, 59, and 61 from section 5.6.

When done, compare your solutions with those in the Student Solutions Manual.

U5-26 Unit 5: Introduction to Trigonometry

TRU Open Learning

5.7: Applied Problems

Introduction

Trigonometry is used to solve applied problems involving triangles or to model

periodic phenomena. In this section we will be restricting our study to geometric

applications involving right triangles.

Section Learning Outcomes

After completing this section you should be able to:

 Solve for all sides and angles of a right triangle.

 Use trigonometric ratios to solve applied problems.

Study Plan

1. Read section 5.7 of the textbook.

2. Read the following “Study Notes and Sample Questions” section to prepare for

the practice exercises.

3. Follow the instructions regarding “Further Practice.”

Study Notes and Sample Questions

Not all of the material presented in each section of the textbook is required for this

course and some topics are more important than others. To master the required

material in this section, review the following topics and examples and the

supplemental “Sample Questions from Section 5.7”.

Section 5.7: Applied Problems

Topic Page Number/Example

Solving the Right Triangle Page 393

Solving a Right Triangle Page 394—Example 1

Solving a Right Triangle Page 395—Example 2

Using an Angle of Elevation Page 395—Example 3

Using Angles of Depression Page 396—Example 4

MATH 1001: Pre-Calculus Mathematics U5-27

TRU Open Learning

Sample Questions from Section 5.7

Page 399—q. 6

Given the indicated parts of a triangle ABC with 90   , find the exact values of the remaining parts:

4 3 8a c 

Solution:

4 3 3 sin

8 2

c     60  

90 60 30     

2 2 64 48 16 4b c a     

Page 400—q. 26

Surveying

From a point 15 meters above level ground, a surveyor measures the angle of

depression of an object on the ground at 68 . Approximate the distance from the object to the point on the ground directly beneath the surveyor.

Solution:

15 15 tan 68 6.1

tan 68 x

x      15

So 6.1x m

U5-28 Unit 5: Introduction to Trigonometry

TRU Open Learning

Further Practice Section 5.7

Check your understanding and improve your speed by working through some of

the exercises on pages 399–400 of the textbook. Do enough of the odd‐numbered

questions of each type to convince yourself that you can get the right answers. Note

that the answers are at the back of the textbook and complete worked‐out solutions

are in the Student Solutions Manual––but try to avoid looking at answers or solutions

until you have made your own best effort.

As a minimum you should do questions 1, 5, and 25 from section 5.7. When done,

compare your solutions with those in the Student Solutions Manual.