Chapter 4Identities

4.1 Fundamental Identities and Their Use

4.2 Verifying Trigonometric Identities

4.3 Sum, Difference, and Cofunction Identities

4.4 Double-Angle and Half-Angle Identities

4.5 Product-Sum and Sum-Product Identities

Fundamental Identities and Their UseFundamental identitiesEvaluating trigonometric identitiesConverting to equivalent forms

Fundamental Identities

Evaluating Trigonometric IdentitiesExample

Find the other four trigonometric functions of x when

cos x = -4/5 and tan x = 3/4

3

5

53

1

sin

1csc

5

3

4

3

5

4))(tan(cossin

3

4

431

tan

1cot

4

5

54

1

cos

1sec

xx

xxx

xx

xx

Simplifying Trigonometric Expressions

xx

x

x

x

x2

2

2

2

2

2tan

cos

sin

cos

cos11

cos

1

xx

22

tan1cos

1

xxx

xx 2cos21cottan

cottan

•Claim:

•Proof:

•Claim:

•Proof:

xxx

xx

xx

xx

xx

xx

xx

xx

xx

xx

xx

xx

xx

xx

xx

222

22

22

cos211

coscos1

cossin

cossin

sincos

cossin

sincos

cossin

)(cossin

)(cossin

sincos

cossin

sincos

cossin

cottan

cottan

4.2 Verifying Trigonometric Identities

Verifying identitiesTesting identities using a graphing

calculator

Verifying Identities

Verify csc(-x) = -csc x

xxxx

x cscsin

1

sin

1

)sin(

1)csc(

Verify tan x sin x + cos x = sec x

xxx

xxxx

x

xxxx sec

cos

1

cos

cossincossin

cos

sincossintan

22

Verifying Identities

Verify right-to-left:x

xxx

cos1

sincotcsc

xx

x

x

xx

x

x

xx

x

xx

xx

xx

x

x

cotcscsin

cos

sin

1

sin

cos1

sin

cos1sin

cos1

cos1sin

cos1cos1

cos1sin

cos1

sin

2

2

Verifying Identities Using a Calculator

xx

xcsc

cos1

sin2

Graph both sides of the equation in the same viewing window. If they produce different graphs they are not identities. If they appear the same the identity must still be verified.

Example:

4.3 Sum, Difference, and Cofunction Identities

Sum and difference identities for cosineCofunction identitiesSum and difference identities for sine and

tangentSummary and use

Sum and Difference Identities for Cosine

cos(x – y) = cos x cos y - sin x sin y Claim: cos(/2 – y) = siny

Proof:cos(/2 – y) = cos (/2) cos y + sin(/2) sin y

= 0 cos y + 1 sin y = sin y

Sum and Difference Formula for Sine and Tangent

sin (x- y) = sin x cos x + cos x sin y

yx

yx

yxyx

yxyx

yxyx

yxyx

yxyx

yxyx

yx

yxyx

tantan1

tantan

coscossinsin

coscoscoscos

coscossincos

coscoscossin

sinsincoscos

sincoscossin

cos

sintan

Finding Exact Values

Find the exact value of cos 15ºSolution:

4

132

22

13

2

1

2

1

2

3

2

1

30sin45sin30cos45cos

)3045cos(15cos

Double-Angle and Half-Angle Identities

Double-angle identitiesHalf-angle identities

Double-Angle Identities

2

2cos1cos and

2

2cos1sin 22 x

xx

x

Using Double-Angle IdentitiesExample:

Find the exact value of cos 2x if sin x = 4/5, /2 < x <

The reference angle is in the second quadrant.

25

7

5

4212cos

3

4tan,

5

4sin

31625

2

x

xx

a

Half-Angle Identities

Using a Half-Angle Identity

Example: Find cos 165º.

2

32

223

1165cos

2

330cos330cos

2

330cos1

2

330cos165cos

4.5 Product-Sum and Sum-Product Identities

Product-sum identitiesSum-product identitiesApplication

Product-Sum Identities

Using Product-Sum Identities

Example: Evaluate sin 105º sin 15º.Solution:

4

1

2

10

2

1120cos90cos

2

1

15105cos15105cos2

115sin105sin

Sum-Product Identities

Using a Sum-Product Identity

Example: Write the difference sin 7 – sin 3 as a product.

Solution:

2sin5cos22

37sin

2

37cos2

3sin7sin