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Chapter 7 Section 7.4 Double-Angle and Half-Angle Identities The angle identities we studied in the last section will generate other identities concerning the angle that are extremely useful in trigonometry. In particular we will see later when it comes to solving trigonometric equations. In the sum of angles formulas we had before if we replace the s and t each by an x we get each of the double angle identities below. sin( x x) sin x cos x cos x sin x 2 sin x cos x sin( 2 x) 2 sin x cos x cos( x x) cos x cos x sin x sin x cos( 2 x) cos x sin x cos 2 x sin 2 x cos 2 x (1 cos 2 x) 2 cos 2 x 1 2(1 sin 2 x) 1 1 2 sin x 2 x tan x tan( x x) 1tan tan x tan x 2 tan x 1 tan 2 x 2 2 2 cos x 1 2 1 2 sin x 2 tan( 2x) 2 tan x 1 tan2 x Example: If x is an angle in quadrant IV with cos x = ⅓ find cos(2x) and sin(2x). cos(2 x) 2 cos x 1 2 2 1 2 3 1 1 2 9 7 9 To apply the double angle formula we need to find sin x first. This will be negative since the angle is in quadrant IV. sin x 1 cos x 1 2 1 2 3 Now we can apply the double angle formula for sine. 1 1 9 8 9 sin( 2 x) 2 sin x cos x 2 8 3 1 3 8 3 2 8 9 These identities can be extended to triple angle (or larger) by applying the sum of angles identity to the angle 3x=2x+x. These identities are not needed very often but it is possible to derive them when you need them. Identities for Lowering Powers The double angle identities for the cos(2x) can be rewritten in a different form to produce identities that reduce the power on either the sine or cosine. Again these are very useful when it comes to solving trigonometric equations. cos( 2 x) 1 2 sin x 2 2 sin 2 x 1 cos( 2 x) sin x 2 1 cos(2 x ) 2 cos( 2 x) 2 cos x 1 2 1 cos( 2 x) 2 cos x 2 cos x 2 1 cos(2 x ) 2 We show how these can be used to reduce the powers on the expression below. 4 sin x sin x 2 2 2 1 cos(2 x ) 2 sin x cos x 2 1 cos(2 x ) 2 2 1 2 cos(2 x ) cos 2 ( 2 x ) 4 4 x) 1 2 cos(2 x ) 1cos( 2 4 2 4 cos(2 x ) 1 cos(4 x ) 8 3 4 cos(2 x ) cos(4 x ) 8 1 cos 2 ( 2 x ) 4 4x) 1 1cos( 2 4 2 (1 cos(4 x ) 8 1 cos(4 x ) 8 1 cos(2 x ) 2 Half-Angle Formulas The following identities can be obtained by replacing x by u/2 in the double angle formulas. The choice of + or – sign in the sine and cosine formulas depend on the quadrant the angle u/2 is in. sin u 2 cos u 2 tan u 2 1 cos u 2 1 cos u 2 sinu 1 cos u Find the exact value of cos(112.5), sin(112.5), and tan(112.5) Since 112.5 is half of 225 we will use the half-angle formula for the cos(225/2). Since 112.5 is in the second quadrant the cosine will be negative and sine positive. cos 112.5 cos 225 2 1 cos 225 2 sin 112.5 sin 225 2 1 cos 225 2 1 2 2 2 1 2 2 2 2 2 4 2 2 4 2 2 2 2 2 2 tan 112.5° = sin 225° 1+cos 225° = 2 2 2 1− 2 = − 2 2− 2 − sin( s t ) sin s cos t cos s sin t sin( s t ) sin s cos t cos s sin t Product-To-Sum and Sum-To-Product Identities: If we take the sum and difference identities and add or subtract them we get ways to turn products into sums or sums into products Product-To-Sum sin s cos t 12 sin( s t ) sin( s t ) sin( s t ) sin( s t ) cos s cos t 12 cos( s t ) cos( s t ) sin s sin t 12 cos( s t ) cos( s t ) cos s sin t 1 2 sin( s t ) sin( s t ) 2 sin s cos t sin s cos t 1 2 sin( s t ) sin( s t ) Sum-To-Product sin s sin t 2 sin s 2t cos s2t sin s sin t 2 cos s 2t sin s2t cos s cos t 2 cos s 2t cos s2t cos s cos t 2 sin s 2t sin s2t cos( 4 x) sin( 7 x) 12 sin( 4 x 7 x) sin( 4 x 7 x) Simplify: cos(4x) sin(7x) 12 sin( 11x) sin( 3x) 12 sin( 11x) sin( 3x)