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Faculty of Science

Unit 6: Analytic Trigonometric

Functions

MATH 1001 Pre-Calculus Mathematics

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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

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Table of Contents Unit 6: Analytic Trigonometric Functions

Introduction ................................................................................................................. U6‐1

Learning Outcomes ..................................................................................................... U6‐1

6.1: Verifying Trigonometric Identities .................................................................. U6‐2

Introduction .............................................................................................................. U6‐2

Section Learning Outcomes .................................................................................... U6‐2

Study Plan ................................................................................................................. U6‐2

Sample Questions from Section 6.1 ....................................................................... U6‐3

Further Practice Section 6.1 ..................................................................................... U6‐6

6.2: Trigonometric Equations .................................................................................... U6‐7

Introduction .............................................................................................................. U6‐7

Section Learning Outcomes .................................................................................... U6‐7

Study Plan ................................................................................................................. U6‐7

Sample Questions from Section 6.2 ....................................................................... U6‐8

Further Practice Section 6.2 ................................................................................... U6‐10

6.3: The Addition and Subtraction Formulas ....................................................... U6‐11

Introduction ............................................................................................................ U6‐11

Section Learning Outcomes .................................................................................. U6‐11

Study Plan ............................................................................................................... U6‐11

Sample Questions for Section 6.3 ......................................................................... U6‐12

Further Practice Section 6.3 ................................................................................... U6‐14

6.4: Multiple‐Angle Formulas ................................................................................. U6‐15

Introduction ............................................................................................................ U6‐15

Section Learning Outcomes .................................................................................. U6‐15

Study Plan ............................................................................................................... U6‐15

Sample Questions for Section 6.4 ......................................................................... U6‐15

Further Practice Section 6.4 ................................................................................... U6‐17

6.6: The Inverse Trigonometric Functions ............................................................ U6‐18

Introduction ............................................................................................................ U6‐18

Section Learning Outcomes .................................................................................. U6‐18

Study Plan ............................................................................................................... U6‐18

Sample Questions from Section 6.6 ..................................................................... U6‐19

Further Practice Section 6.6 ................................................................................... U6‐21

MATH 1001: Pre-Calculus Mathematics U6-1

TRU Open Learning

Unit 6: Analytic Trigonometric Functions This unit corresponds to Chapter 6 of the textbook. In Unit 6 you will be covering

the following sections from your textbook:

 6.1: Verifying Trigonometric Identities

 6.2: Trigonometric Equations

 6.3: The Addition and Subtraction Formulas

 6.4: Multiple‐Angle Formulas

 6.6: The Inverse Trigonometric Functions

Introduction In advanced mathematics it is sometimes necessary to simplify a trigonometric

expression by using an identity or formula. In particular, solving trigonometric

equations often requires algebraic methods that involve identities. In the last section

of this unit you will be introduced to the inverse trigonometric functions—another

tool to solve trigonometric equations.

Learning Outcomes In this unit you will learn how to verify identities and use them to simplify

trigonometric expressions. You will also learn how to use them to solve

trigonometric equations. Specific learning outcomes are listed at the beginning of

each section in the unit.

U6-2 Unit 6: Analytic Trigonometric Functions

TRU Open Learning

6.1: Verifying Trigonometric Identities

Introduction

After working hard to solve an algebra problem you check the answer and it looks

different from yours. But, after a moment of panic, you realize that it’s really just

saying the same thing in a different way. This can often be seen by means of an

algebraic identity (i.e. an equation, like 2 22 1 ( 1)x x x    , which is true for all values of the variable). You also frequently use identities to simplify expressions and

to rearrange the terms in an equation to make it easier to find a solution.

Identities involving trigonometric functions are important for similar reasons. You

have already seen some examples, and in this section you will find more. You will

also learn how to work with identities you know in order to get new ones.

Section Learning Outcomes

After completing this section you should be able to:

 Use the Pythagorean, reciprocal, tangent, and cotangent identities to verify other identities.

Study Plan

1. Read section 6.1 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 6.1”.

Section 6.1: Verifying Trigonometric Identities

Topic Page, Example Number

Trigonometric Expressions Page 416—Illustration

Verifying an Identity Page 416—Example 1

Page 418—Examples 2, 3

Page 419—Example 4

MATH 1001: Pre-Calculus Mathematics U6-3

TRU Open Learning

Sample Questions from Section 6.1

Page 421—q. 16

Verify the identities:

1 csc cot

csc cot y y

y y  



Solution:

2 2

1 1 csc cot

csc cot csc cot csc cot

csc cot

csc cot 1 cos

sin sin

csc cot 1 cos

sin

csc cot sin sin

csc cot

y y

y y y y y y

y y

y y y

y y

y y y

y y y y

y y

  

  

 



 



 



 

 

Page 421—q. 34

tan tan cot cot

1 tan tan cot cot 1

u v v u

u v u v

  

 

U6-4 Unit 6: Analytic Trigonometric Functions

TRU Open Learning

Solution:

1 1 tan tan cot cot

1 11 tan tan 1 cot cot

cot cot cot cot

cot cot 1 cot cot

cot cot

cot cot 1

u v u v u v

u v

v u u v

u v u v

v u

u v

 

  



 

 



Page 421—q. 38

2 2cot tan csc sec sin cos

y y y y

y y

  

Solution:

2 2

2 2 2 2

2 2

cos sin cot tan sin cos sin cos sin cos

cos sin 1

sin cos sin cos

cos sin

sin cos

cos sin

sin cos sin cos

1 1

sin cos

csc sec

y y y y y y y y y y

y y

y y y y

y y

y y y y

y y

 



  

 

 

MATH 1001: Pre-Calculus Mathematics U6-5

TRU Open Learning

Page 421—q. 48

   ln sec ln cos  

Solution:

         1ln sec ln ln 1 ln cos 0 ln cos ln cos cos

    

          

Page 421—q. 50

ln csc cot ln csc cotx x x x   

Solution:

  

2 2

2 2 2

csc cot csc cot ln csc cot ln

csc cot

csc cot ln

csc cot

1 cos 1 cos sin sin sinln ln csc cot csc cot

sin sinln

csc cot

1 ln

csc cot

x x x x x x

x x

x x x x x x x x x

x x

   



     

       

         

      

   

    

  

      

 

1 ln csc cot

ln csc cot

x x

  

  

Page 422—q. 52

Show that the equation is not an identity. (Hint: Find one number for which the

equation is false.):

2 2sin cos sin cost t t t  

U6-6 Unit 6: Analytic Trigonometric Functions

TRU Open Learning

Solution:

For t  ,

   

22 2sin cos 0 1 1 1

sin cos 0 1 1

LHS

RHS

 

      

      

1 1

LHS RHS

    

Since this equation is not true for all real values of t (in this case, it is false for t  ) it follows that this is not an identity.

Further Practice Section 6.1

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

the exercises on pages 420–421 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, 11, 13, 15, 17, 29, 37, and 45 from

section 6.1. When done, compare your solutions with those in the Student Solutions

Manual.

MATH 1001: Pre-Calculus Mathematics U6-7

TRU Open Learning

6.2: Trigonometric Equations

Introduction

In Unit 5 we briefly addressed the problem of finding angles from given

trigonometric ratios. This amounts to solving for t in an equation like sin( )t v where v is a given value. In this section, we shall look at such problems in more

detail and will also find angles from more complicated equations involving their

trigonometric ratios.

Section Learning Outcomes

After completing this section you should be able to:

 Solve simple trigonometric equations by recognizing particular special cases.

 Use identities and algebraic manipulation to solve more complicated equations.

Study Plan

1. Read section 6.2 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 6.2”.

U6-8 Unit 6: Analytic Trigonometric Functions

TRU Open Learning

Section 6.2: Trigonometric Equations

Topic Page, Example Number

Definition of a Trigonometric Equation Page 422

Solving a Trigonometric Equation

Involving the Sine Function

Page 423—Example 1

Solving a Trigonometric Equation

Involving the Tangent Function

Page 423—Example 2

Solving a Trigonometric Equation

Involving Multiple Angles

Page 424—Example 3

Solving a Trigonometric Equation by

Factoring

Page 425—Example 4

Page 426—Example 5

Page 427—Example 6

Sample Questions from Section 6.2

Page 432—q. 2, 14, 17

Find all solutions of the equation:

q. 2

cos 1t  

Solution:

1cos 1 cos ( 1) 2t t n      where 0, 1, 2,n    

2t n    or  2 1t n  

q. 14

1 2 cos

4 2 x    

 

MATH 1001: Pre-Calculus Mathematics U6-9

TRU Open Learning

Solution:

Since cosine is negative, angle 4

terminates in quadrant II or III. And thus

QII : 2 or QIII : 2 4 4 4 4ref ref

x x x x n n              

   

First determine the reference angle as follows: 2

cos 4 2 4 4ref ref

x x                 

QII : 2 or QIII : 2 4 4 4 4

x x x x n n        

3 5 2 or 2

4 4 4 4

x x n n

      

3 8 or 5 8x n x n      

q. 17

1 sin 2

3 2 x

     

Solution:

11 1sin 2 2 sin 2 3 2 3 2

x x n  

               

; 0, 1, 2,n    

Sine is positive in Q I and Q II, and so

Q I: 2 2 2 2 3 6 2 4

x n x n x n    

          

Q II: 5 7 7

2 2 2 2 3 6 6 12

x n x n x n    

          

for 0, 1, 2,n    

U6-10 Unit 6: Analytic Trigonometric Functions

TRU Open Learning

Page 433—q. 48

Find the solutions of the equation that are in the interval  0, 2 . 2cot cot 0  

Solution:

 2cot cot 0 cot cot 1 0 cot 0 or cot 1 0

3 5 ; or cot 1 ;

2 2 4 4

         

  

    

   

    

Page 433—q. 76

Find the solutions of the equation that are in the interval  0, 2 . 25cos 3cos 2 0   

Solution:

  

 

1 1

5 cos 3cos 2 0

5cos 2 cos 1 0 5 cos 2 0 cos 1 0

2 cos cos 1

5 2

cos cos 1 5

     

 

  

  

       

   

       

Cosine is positive in Q I and Q IV and so

1.1593 2 1.1593 5.1239 3.1416or or          

Further Practice Section 6.2

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

the exercises on pages 432–433 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, 7, 9, 11, 25, 43, and 75 from section 6.2.

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

MATH 1001: Pre-Calculus Mathematics U6-11

TRU Open Learning

6.3: The Addition and Subtraction Formulas

Introduction

In this section you will learn the formulas for the sum and differences of real

numbers or angles and use them to develop further identities and to solve applied

problems.

Section Learning Outcomes

After completing this section you should be able to:

 Use the addition and subtraction formulas to prove other identities and formulas.

 Use the addition and subtraction formulas to find the trigonometric function values of the sum and difference of angles or real numbers.

Study Plan

1. Read section 6.3 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 6.3”.

Section 6.3: The Addition and Subtraction Formulas

Topic Page, Example Number

Subtraction Formula for Cosine (No Proof) Page 436

Using a Subtraction Formula Page 437—Example 1

Addition Formula for Cosine Page 438

Using an Addition Formula Page 438—Example 2

Addition and Subtraction Formulas for

Sine (No Proof)

Page 439

U6-12 Unit 6: Analytic Trigonometric Functions

TRU Open Learning

Sample Questions for Section 6.3

Find the exact values:

(a) cos135 cos 60   (b)  cos 75 use 75 135 60     

Solution:

(a) cos135 cos 60  

cos 45 cos 60

2 1

2 2

2 1

    

  

  

(b)  cos 75 cos 135 60    

cos135 cos 60 sin135 sin 60

2 1 2 3

2 2 2 2

2 6

 

                  

  

   

Page 443—q. 14

Express a trigonometric function of one angle:

sin 57 cos 4 cos 57 sin 4    

Solution:

 sin 57 cos 4 cos 57 sin 4 sin 57 4 sin 61         

Page 444—q. 24a

If  and  are acute angles such that 13csc 12

  and 4

cot 3

  , find:

 sin   Solution:

 sin sin cos cos sin       

13 12 csc sin

12 13    

2 2 2 12 25 5cos sin 1 cos 1

13 169 13               

 

MATH 1001: Pre-Calculus Mathematics U6-13

TRU Open Learning

Since  and  are acute, choose the positive value. So 5cos 13

 

4 3 cot tan

3 4    

2 2 2 3 25 5sec 1 tan sec 1

4 16 4               

 

Again, choose the positive value. So 5 4

sec cos 4 5

   

Finally,   12 4 5 3 63sin 13 5 13 5 65

      

Page 444—q. 30

Verify the reduction formula:

sin cos 2

x x    

 

Solution:

   

sin 2

sin cos cos sin 2 2

sin 0 cos 1

cos

LHS x

x x

x RHS



 

     

 

 

Page 444—q. 42

Verify the identity

 2cos cos sin 4 2

         

U6-14 Unit 6: Analytic Trigonometric Functions

TRU Open Learning

Solution:

 

cos cos cos sin sin 4 4 4

2 2 cos sin

2 2

2 cos sin

LHS

RHS

     

 

                   

        

   

 



Page 444—q. 50

Verify the identity:

  1 cos cos

tan tan sin

     

  

Solution:

 

1 1 1 sin sin sin cos cos sintan tan cos cos cos cos

cos cos

sin

LHS

RHS

          

   

    

  

Further Practice Section 6.3

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

the exercises on pages 443–444 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, 11, 13, 15, 17, 19, 21 (a), 23 (a), 25, 27,

29, and 37 from section 6.3. When done, compare your solutions with those in the

Student Solutions Manual.

MATH 1001: Pre-Calculus Mathematics U6-15

TRU Open Learning

6.4: Multiple-Angle Formulas

Introduction

In this section you will learn the double angle formula and use it in a variety of

applications.

Section Learning Outcomes

After completing this section you will be able to:

 Use the double angle formula to calculate function values.

 Use the double angle formula to simplify trigonometric expressions and verify identities.

Study Plan

1. Read section 6.4 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 6.4”.

Section 6.4: Multiple Angle Formulas

Topic Page, Example Number

Double Angle Formulas (1) and (2) Page 446

Using Double Angle Formulas Page 447—Example 1

Sample Questions for Section 6.4

Find the exact values of sin 2 , cos 2  and tan 2 for the given values of  :

4 sin

5    270 360  

U6-16 Unit 6: Analytic Trigonometric Functions

TRU Open Learning

Solution:

4 sin

5    and since 2 2sin cos 1   we have

2 24 16 9 3cos 1 cos 1

5 25 25 5              

 

Since  terminates in Q IV, cosine is positive and so 3

cos 5

 

Thus:

sin 2 2 sin cos

4 3 2

5 5

  

        

 

2 2

cos 2 cos sin

3 4 cos 2

5 5

9 16 cos 2

25 25 7

cos 2 25

  



 

            

 

sin 2 tan 2

cos 2 24 7

25 25

 

 

             



Page 453—q. 44

Find the solutions of the equation that are in the interval  0, 2 :

cos sin 2 0t t 

Solution:

  cos sin 2 0 cos 2 sin cos 0

cos 1 2 sin 0

1 cos 0 or sin

2 3 5

, or , 2 2 6 6

t t t t t

t t

t t    

    

  

  

MATH 1001: Pre-Calculus Mathematics U6-17

TRU Open Learning

Page 454—q. 60

Projectile’s Range

If a projectile is fired from ground level with an initial velocity of v ft/sec and at an

angle of  degrees with the horizontal, the range R of the projectile is given by: 2

sin cos 16

v R  

If 80 ft / secv  , approximate the angles that result in a range of 150 feet.

Solution:

280 150, 80 150 sin cos

16 3

sin cos 8

3 3 2 2 sin cos sin 2

8 4

R v  

 

  

   



       

So  12 sin 0.75 48.59 or 180 48.59 131.41       

And then 24.3 or 65.7   

Further Practice Section 6.4

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

the exercises on pages 452–453 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, 6, 23, and 43 from section 6.4. When

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

U6-18 Unit 6: Analytic Trigonometric Functions

TRU Open Learning

6.6: The Inverse Trigonometric Functions

Introduction

In section 4.1 of Unit 4 we learned how to find the inverse of a function. In this

section we will discuss the inverses for the sine, cosine and tangent functions.

Section Learning Outcomes

After completing this section you should be able to:

 Find the exact values of the inverse trigonometric functions of special real numbers.

 Sketch the graph of the inverse sine, cosine and tangent functions.

 Find the exact values of composite functions involving trigonometric and inverse trigonometric functions.

 Solve trigonometric equations using inverse trigonometric functions.

Study Plan

1. Read section 6.6 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 6.6”.

MATH 1001: Pre-Calculus Mathematics U6-19

TRU Open Learning

Section 6.6: The Inverse of Trigonometric Functions

Topic Page, Example Number

Relationships between 1f  and f Pages 460, 461

Definition of the Inverse Sine Function Page 461

Warning Page 461

Properties of 1sin Page 462

Using Properties of 1sin Page 462—Example 1

Finding a Value of 1sin Page 463—Example 2

Definition of the Inverse Cosine

Function

Page 463

Warning Page 464

Properties of 1cos Page 465

Finding a Trigonometric Function Value Page 465—Example 4

Definition of the Inverse Tangent

Function

Page 465

Using Inverse Trigonometric Functions

to Solve an Equation

Page 468—Example 9

Sample Questions from Section 6.6

Page 471—q. 12

Find the exact value of the expression whenever it is defined:

a) 1 2

sin sin 3

      

b) 1 4

cos cos 3

      

c) 1 7

tan tan 6

      

U6-20 Unit 6: Analytic Trigonometric Functions

TRU Open Learning

Solution:

a) 1 1 2 3

sin sin sin 3 2 3

             

b) 1 1 4 1 2

cos cos cos 3 2 3

              

c) 1 7

tan tan 6

      

1 3tan 3 6

      

Page 471—q. 14

Find the exact value of the expression whenever it is defined:

a)  1sin tan 3 b)  1cos sin 1 c)  1tan cos 0 Solution:

a)  1 3sin tan 3 sin 3 2

      

b)  1cos sin 1 cos 0 2

      

c)  1tan cos 0 tan 2

      

which is undefined

Page 473—q. 66, 70

Use inverse trigonometric functions to find the solutions of the equations that are in

the given interval, and approximate the solutions to four decimal places.

q. 66

4 23 tan 19 tan 2 0 , 2 2

         

 

Solution:

   22 23 tan 19 tan 2 0    Let tanx 

MATH 1001: Pre-Calculus Mathematics U6-21

TRU Open Learning

3 19 2 0

19 361 24 19 337 19 337 tan tan

6 6 6

x x

x  

  

         

1 1

19 337 19 337 tan tan 1.1896

6 6

19 337 tan 0.3162

6 or

 



 



                      

          

q. 70

26sin sin 2x x  ; 0, 2 Solution:

26sin sin 2 0x x  

   2 13sin 2 2 sin 1 0 sin or sin 3 2

x x x x      

1 2sin 0.7297 3

      

and since sine is positive in Q I and QII

0.7297 0.7297 2.4119orx x    

1 1sin 2 6

      

and since sine is negative in QIII and Q IV

7 11 3.6652 5.7596

6 6 orx x

     

Further Practice Section 6.6

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

the exercises on pages 471–473 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 bare minimum you should do questions 1, 3, 5, 9, 13, 61, and 63 from section

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