14.7 Apply Double-Angle and Half-Angle Formulas

Goal · Use double-angle and half-angle formulas.

Your Notes

DOUBLE-ANGLE AND HALF-ANGLE FORMULAS

Double-Angle Formulas	Half-Angle Formulas
cos 2a = __cos^2 a__ - __sin^2 a__
cos 2a = __2 cos^2 a__ - __1__
cos 2a = __1__ - __2 sin^2 a__
Sin 2a = __2 sin a cos a__
tan 2a = ___________

Example 1

Evaluating trigonometric expressions

Find the exact value of cos .

Solution

Because is in Quadrant I and the value of cosine is positive in Quadrant I, the following formula is used:

Cos =

_{_________}

Your Notes

Example 2

Evaluate trigonometric expressions

Given sin a = with < a < p, find (a) cos 2a and (b) cos .

Solution

Using a Pythagorean identity gives cos a = _{____.}

_a. cos 2a = __2 cos² a__ - 1 = __2__ _{______} - 1 = _{____}

b. Because is in Quadrant I, cos is ___positive_ .

Checkpoint Complete the following exercises.

1. Find the exact value of sin .

2. Given cos a = - with p < a < , find sin 2a.

Example 3

Verify a trigonometric identity

Verify the identity sin 4x = 4 sin x cos x(l - 2 sin² x)

Solution

sin 4x = sin(2x + __2x__)

= sin 2x cos 2x + ___cos2x sin2x__

= (2 sin x cos x) cos 2x + cos 2x(__2 sin x cos x___)

= (2 sin x cos x + 2 sin x cos x) ___cos 2x__

= (4 sin x cos x) ___cos 2x__

= 4 sin x cos x(___1 - 2 sin² x___)

Your Notes

Example 4

Solve a trigonometric equation

Solve cos 2x + sin x = 0 for 0 < x < 2p.

Solution

cos 2x + sin x = 0

___1 - 2 sin² x + sin x__ = 0

___- 2 sin² x__ + sin x + __1__ = 0

( ___2 sin x__ + __1__ )( ___-sin x__ + __1__) = 0

__2 sin x__ + ___1__ = 0 or	__-sin x___ + __1__ =0
__2 sin x__ = -1	__-sin x__ = -1
sin x = ______	sin x = ___1__
x = _____, ______	x = ______

Checkpoint Complete the following exercises.

3. Verify the identity cos 3x = cos³ x - 3 sin² x cos x.

cos 3x= cos(2x + x)

= cos 2x cos x - sin 2x sin x

= (cos² x - sin² x) cos x - (2 sin x cos x) sin x

= cos³ x - sin² x cos x - 2 sin² x cos x

= cos³ x - 3 sin² x cos x

4. Solve tan = sin x for 0 £ x < 2p.

0,