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Multiple-Angle Formulas (Sec. 7.4) Feature Details / Examples Multiple angle formulas Formulas for trigonometric functions of multiples of an angle Double-angle Formulas Formulas for trigonometric functions of twice an angle in terms of functions of the basic angle. 1. sin 2u = 2 sin u cos u 2. (a) cos 2u = cos2 u – sin2 u (b) cos 2u = 1 – 2 sin2 u (c) cos 2u = 2 cos2 u – 1 3. tan 2u = 2 tan u / (1 – tan2 u) Example 1: Using doubleangle formulas If sin α = 4/5 and α is an acute angle, find the exact values of sin 2α and cos 2α. Half-Angle Identities 1. sin 2 u = 1 − cos 2u 2 2. cos 2 u = 1 + cos 2u 2 3. tan2 u = 1 − cos 2u 1 + cos 2u € € Example 2: Using half- € angle identities to verify an identity Verify the identity sin2 x cos2 x = 1/8(1–cos 4x) . 1 Half-Angle Formulas v 1 − cos v sin = ± 2 2 v 1 + cos v cos = ± 2 2 € € Half-Angle Formulas vs. Half-Angle Identities € Half-Angle Formulas for Tangent Example 3 € v 1 − cos v tan = ± 2 1 + cos v Why are some of these called “formulas” while others are called “identities”? Obviously the “formulas” aren’t identities, right, Mr. Byrd? __________. 1. tan v 1 − cos v = 2 sin v 2. tan v sin v = 2 1 + cos v € DAB, April 2011 2