Chapter 11: Trigonometric Identities and Equations

11.1 Trigonometric Identities

11.2 Addition and Subtraction Formulas

11.3 Double-Angle, Half-Angle, and Product-Sum Formulas

11.4 Inverse Trigonometric Functions

11.5 Trigonometric Equations

11.3 Double-Angle, Half-Angle, and Product-Sum Formulas

Double-Angle IdentitiesE.g. cos 2A = cos(A + A)

= cos A cos A – sin A sin A = cos² A – sin² A

Other forms for cos 2A are obtained by substituting either cos² A = 1 – sin² A or sin² A = 1 – cos² A to get

cos 2A = 1 – 2 sin² A or cos 2A = 2 cos² A – 1.

AA

A 2tan1tan2

2tan

1cos22cos

sincos2cos2

22

AA

AAA

AAA

AA

cossin22sin

sin212cos 2

11.3 Finding Function Values of 2ExampleGiven and sin < 0, find a) sin 2, b) cos 2, and c) tan 2.

SolutionTo find sin 2, we must find sin .

53cos

Choose the negative square root since sin < 0.

25

91sin

cossin22sin) a

5

3

5

42

25

24

15

3sin

22

1cossin 22

5

4sin

So is…

None of t

he a...

0% 3% 3%

94%

0%

2cos

1. 2. 3. 4. 5. None of the above

25

77

245

45

8

11.3 Finding Function Values of 2

OR 7

24

1

22

34

34

22

5

4

5

3

22 sincos2cos ) b

25

7

3

4

cos

sin tan where,

tan1

tan22tan )

53

54

2

c

2cos

2sin2tan

257

2524

7

24

11.3 Simplifying Expressions Using Double-NumberIdentities

Example Simplify each expression.

(a) cos² 7x – sin² 7x (b) sin 15° cos 15°

Solution(b) cos 2A = cos² A – sin² A. Substituting 7x in for A

gives cos² 7x – sin² 7x = cos 2(7x) = cos 14x.

(c) Apply sin 2A = 2 sin A cos A directly.

)152sin(2

1 4

130sin

2

1

15cos15sin)2(2

115cos15sin

11.3 Product-to-Sum Identities• Product-to-sum identities are used in calculus to find

integrals of functions that are products of trigonometric functions.

• Adding identities for cos(A + B) and cos(A – B) gives

)].cos()[cos(21

coscos

coscos2)cos()cos(

sinsincoscos)cos(

sinsincoscos)cos(

BABABA

BABABA

BABABA

BABABA

11.3 Product-to-Sum Identities

• Similarly, subtracting and adding the sum and difference identities of sine and cosine, we may derive the identities in the following table.

Product-to-Sum Identities

)]sin()[sin(sincos

)]sin()[sin(cossin

)]cos()[cos(sinsin

)]cos()[cos(coscos

21

21

21

21

BABABA

BABABA

BABABA

BABABA

11.3 Using a Product-to-Sum IdentityExampleRewrite cos 2 sin as the sum or difference of two

functions.

SolutionBy the identity for cos A sin A, with 2 = A and = B,

)]sin()[sin(sincos Since, 21 BABABA

.sin2

13sin

2

1

)]2sin()2[sin(2

1

sin2cos

11.3 Sum-to-Product Identities

• From the previous identities, we can derive another group of identities that are used to rewrite sums of trigonometric functions as products.

Sum-to-Product Identities

22

22

22

22

sinsin2coscos

coscos2coscos

sincos2sinsin

cossin2sinsin

BABA

BABA

BABA

BABA

BA

BA

BA

BA

11.3 Using a Sum-to-Product IdentityExampleWrite sin 2 – sin 4 as a product of two functions.

SolutionUse the identity for sin A – sin B, with 2 = A and 4 = B.

sin3cos2

2

2sin

2

6cos2

2

42sin

2

42cos2

4sin2sin

)sin(3cos2

22 sincos2sinsin Since, BABABA

function oddan is sine Since

11.3 Half-Number Identities• Half-angle identities for sine and cosine are used in calculus when

eliminating the xy-term from an equation of the form Ax² + Bxy + Cy² + Dx + Ey + F = 0, so the type of conic it represents can be determined.

• From the alternative forms of the identity for cos 2A, we can derive three additional identities, e.g.

xx 2sin212cos .sin

2A

Choose the sign ± depending on the quadrant of the angle A/2.

xx 2cos1sin2 2

22cos1

sinx

x .

2 that so 2Let AxAx

2cos1

2sin

AA

11.3 Half-Number Identities

Half-Number Identities

2cos1

2sin

2cos1

2cos

AAAA

AAA

AAA

AAA

sincos1

2tan

cos1sin

2tan

cos1cos1

2tan

11.3 Using a Half-Number Identity to Find an Exact Value

ExampleFind the exact value of

Solution

.12

cos

12cos

26

cos1

26cos

223

1

22

223

1

2

32

11.3 Finding Function Values of x/2

ExampleGiven

SolutionThe half-angle terminates in quadrant II since

.tan and ,sin,cos 222xxx

find ,2 with ,cos 23

32 xx

5

5

.2 243

23 xx

6

30

6

5

2

1 32

2

cos)x

b

6

6

6

1

2

1 32

2sin)x

a

630

66

2

2

cos

sinx

x

2tan)x

c