14.6 Apply Sum and Difference Formulas

Goal · Use trigonometric sum and difference formulas.

Your Notes

SUM AND DIFFERENCE FORMULAS

Sum Formulas

sin(a + b) = _sin a cos b_ + _cos a sin b_

cos(a + b) = _cos a cos b_ - _sin a sin b_

tan(a + b) =

Difference Formulas

sin(a - b) = _sin a cos b_ - _cos a sin b_

cos(a - b) = _cos a cos b_ + _sin a sin b_

tan(a - b) =

Example 1

Evaluate a trigonometric expression

Find the exact value of cos 75°.

Solution

cos 75° = cos(45° + _30°_)	Substitute.
= cos 45° _cos 30°_ - sin 45° _sin 30°_	Sum formula
=	Evaluate.
=	Simplify.

Your Notes

Example 2

Use a difference formula

Find sin(a - b) given that sin a = with < a < 2p

and cos b = with 0 < b < .

Solution

Using a Pythagorean identity and quadrant signs gives

cos a = and sin b = .

sin(a - b) = _sin a cos b_ - _cos a sin b_	Difference formula
=	Substitute
=	Simplify

Example 3

Simplify an expression

Simplify the expression tan(x + p).

Solution

tan (x+ p) =	Sum formula
=	Evaluate.
= _tan x_	Simplify.

Your Notes

Checkpoint Complete the following exercises.

1. Find the exact value of tan .

2. Find cos(a – b) given that sin a = with 0 < a <

and cos b = with < b < .

3. Simplify the expression sin(x + 4p).

sin x

Your Notes

Example 4

Solve a trigonometric equation

Solve cos(x - ) + cos(x + ) = 1 for < x < 2p.

Solution

Use sum and difference formulas to rewrite the original equation as:

p p p
	Simplify the equation.
	Evaluate the cosine function.
	Solve for the cos x

In the interval < x < 2p, the only solution is .

Checkpoint Complete the following exercise.

4. Solve for 0 < x < 2p.

,