MATH 107

Section 6.2

Sum and Difference Formulas

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SUM AND DIFFERENCE FORMULASFOR COSINE

cos cos cos sin sin

cos cos cos sin sin

u v u v u v

u v u v u v

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EXAMPLE 1 Using the Difference Identity for Cosine

Find the exact value of by using

Solution

cos cos12 3 4

12cos

Example

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Find the exact value of cos15 .

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SUM AND DIFFERENCE IDENTITIESFOR SINE

vuvuvu

vuvuvu

sincoscossinsin

sincoscossinsin

19Find the exact value of sin .

12

(Answer on next slide.)

19Find the exact value of sin .

12

19 16 3 4sin sin sin

12 12 12 3 4

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EXAMPLE 5 Using the Sum Formula for Sine

Find the exact value of

without using a calculator.

sin 63ºcos27º cos63ºsin 27º sin 63º 27º

sin 90º

Solution

This expression is the right side of the sum identity for sin (u + v), where u = 63º and v = 27º.

Example

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.sin2

cos vv

BASIC COFUNCTION IDENTITIES

If v is any real number or angle measured in radians, then

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

.cos2

sin vv

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EXAMPLE 3 Using Cofunction Identities

Prove that for any real number x, tan cot .2

x x

Solution

cos

sin

x

x

cot x

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SUM AND DIFFERENCE IDENTITIESFOR TANGENT

tan tantan

1 tan tan

tan tantan

1 tan tan

u vu v

u v

u vu v

u v

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EXAMPLE 11 Verifying an Identity

Verify the identity .

Solution

Apply the difference identity.

tan tantan

1 tan tan

xx

x

0 tantan

1 0 tan

xx

x

tan tanx x

Therefore, the given equation is an identity.