Chapter 4 Analytic Trigonometry Section 4.3 Double-Angle, Half-Angle and Product- Sum Formulas

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.

Example: If x is an angle in quadrant IV with cos x = ⅓ find cos(2 x ) and sin(2 x ). To apply the double angle formula we need to find sin x first. Now we can apply the double angle formula for sine. These identities can be extended to triple angle (or larger) by applying the sum of angles identity to the angle 3 x =2 x + 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(2 x ) 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. We show how these can be used to reduce the powers on the expression below.

Half-Angle Formulas The following identities can 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. Find the exact value of cos(112.5  ) Since  is half of 225  we will use the half-angle formula for the cos(225  /2). Since  is in the second quadrant the cosine will be negative.

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-SumSum-To-Product Simplify: cos(4 x ) sin(7 x )