Chapter 6

Trigonometric Identities and Equations

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SECTION 6.2

Sum and Difference Identities OBJECTIVES 1 2 3 4 Use the sum and difference identities for cosine.

Use the cofunction identities.

Use the sum and difference identities for sine.

Use the sum and difference identities for tangent.

SUM AND DIFFERENCE FORMULAS FOR COSINE cos 

u



v

  cos 

u



v

 

v



v



v v

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3

EXAMPLE 1 Using the Difference Identity for Cosine  Find the exact value of by using  12   3   4 .

cos 12 Solution  cos  12 cos 3 4  cos  3 cos  4  sin  3 sin  4  1 2   4 2 2 2   4 6 2  3  2 2 2  4 6 © 2011 Pearson Education, Inc. All rights reserved

4

BASIC COFUNCTION IDENTITIES If

v

is any real number or angle measured in radians, then cos   2

v

 sin

v

.

sin   2

v

 cos

v

.

If angle

v

is measured in degrees, then replace by 90º in these identities.

 2 © 2011 Pearson Education, Inc. All rights reserved

5

EXAMPLE 3 Using Cofunction Identities Prove that for any real number

x

, tan  2 

x

Solution tan    2 

x

   sin   cos    2  2  

x

 

x

   cos

x

sin

x

 cot

x

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6

SUM AND DIFFERENCE IDENTITIES FOR SINE sin 

u



v

  sin

u

cos

v

 cos

u

sin

v

sin 

u



v

  sin

u

cos

v

 cos

u

sin

v

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7

EXAMPLE 5 Using the Sum Formula for Sine Find the exact value of sin63º cos27º  cos63º sin27º without using a calculator.

Solution This expression is the right side of the sum identity for sin (

u

+

v

), where

u

= 63º and

v

= 27º.     1

8

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EXAMPLE 6 Finding the Exact Value of a Sum Let sin

u

and 3 2 

< v

 3

v

12 = , with

π < u

5 13 < 2

π

. Find the exact value of sin (

u

+

v

).

3  < 2 Solution Find cos

u

.

In QIII, cos < 0.

cos

u

si n 2

u

9 2 5   16 25 cos

u

  4 © 2011 Pearson Education, Inc. All rights reserved 5

9

EXAMPLE 6 Finding the Exact Value of a Sum Solution continued Find sin

v

.

In QIV, sin < 0.

sin

v

sin

v

  5 13 cos 2

v

144 169   25 169 © 2011 Pearson Education, Inc. All rights reserved

10

EXAMPLE 6 Finding the Exact Value of a Sum Solution continued sin (

u

+

v

) = sin

u

cos

v

+ cos

u

sin

v

   3 12 5  13 4 5   5 13      36 65  20 65   16 65 The exact value of sin (

u

 16 +

v

) is . 65 © 2011 Pearson Education, Inc. All rights reserved

11

REDUCTION FORMULA If (

a

,

b

) is any point on the terminal side of an angle  (radians) in standard position, then

a

sin

x



b

cos

x



a

2 

b

2 sin 

x

   for any real number

x

.

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12

EXAMPLE 8 Using the Reduction Formula Find an angle  , in radians, and a real number

A

such that sin

x

 3 cos

x



A

sin 

x

   .

Solution By the reduction formula sin

x

 where

A

3 cos 

x a

2  

a

sin

b

2 

x

 1 2

b

cos

x

   2

A

sin 

x

 4    2  , and  is any angle in standard position that has has the point     1,  © 2011 Pearson Education, Inc. All rights reserved

13

EXAMPLE 8 Using the Reduction Formula Solution continued One such angle is   tan  1    3 .

Then sin

x

 3 cos

x

 2sin 

x

  2 sin  

x

  3   .

 3     © 2011 Pearson Education, Inc. All rights reserved

14

SUM AND DIFFERENCE IDENTITIES FOR TANGENT tan 

u



v

  tan

u

 tan

v v

tan 

u



v

  tan

u

 tan

v v

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15

EXAMPLE 11 Verifying an Identity  

x

   tan

x

.

Solution Apply the difference identity. tan tan      

x

 

x

  tan    tan tan

x x

0  tan

x x

tan   

x

   tan

x

Therefore, the given equation is an identity.

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