Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Double-Angle and Half-Angle Formulas Double-Angle Identities sin 2θ θ = 2 sinθ θ cosθ θ cos 2θ θ = cos2θ Š sin2θ = 1 Š 2 sin2θ = 2 cos2θ Š 1 2tanθ tan 2θ θ = 1 − tan 2θ Three Forms of the DoubleAngle Formula for cos2θ cos 2θ = cos 2 θ − sin 2 θ cos 2θ = 2 cos 2 θ − 1 cos 2θ = 1 − 2 sin 2 θ 1 Power-Reducing Formulas 1 − cos 2θ 2 1 + cos 2θ cos 2 θ = 2 1 − cos 2θ tan 2 θ = 1 + cos 2θ sin 2 θ = Example • Write an equivalent expression for sin4x that does not contain powers of trigonometric functions greater than 1. Solution 1 − cos 2 x 1 − cos 2 x sin 4 x = sin 2 x sin 2 x = 2 2 1 + cos 2 x 1 − 2 cos 2 x + cos 2 2 x 1 − 2 cos 2 x + 2 = 4 4 2 − 4 cos 2 x + 1 + cos 2 x (3 − 3 cos 2 x ) = = 8 8 Half-Angle Identities x sin 2 = ± 1 – cos x 2 x cos 2 = ± 1 + cos x 2 x tan 2 = ± 1 – cos x sin x 1 – cos x 1 + cos x = 1 + cos x = sin x x where the sign is determined by the quadrant in which 2 lies. 2 Text Example Find the exact value of cos 112.5°. Solution Because 112.5° = 225°/2, we use the half−angle formula for cos α/2 with α = 225°. What sign should we use when we apply the formula? Because 112.5° lies in quadrant II, where only the sine and cosecant are positive, cos 112.5° < 0. Thus, we use the − sign in the half−angle formula. cos112.5o = cos 225 2 1 + cos225o =− = 2 =− o − 2 1+ 2 2 2− 2 2− 2 =− 4 2 Half-Angle Formulas for: 1 − cos α 2 sin α sin α α tan = 2 1 + cos α tan α = Example • Verify the following identity: (sin θ − cosθ ) 2 = 1 − sin 2θ Solution (sin θ − cosθ ) 2 = sin 2 θ − 2 sin θ cosθ + cos 2 θ 1 − cos 2θ 1 + cos 2θ + − 2 sin θ cosθ 2 2 2 = − 2 sin θ cosθ = 1 − sin 2θ 2 = 3 Double-Angle and Half-Angle Formulas 4