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5.5 – Multiple Angle and Product to Sum Identities Solving a Multiple Angle Identity cos 2 x cos x 0 2 cos 2 x 1 cos x 0 2 cos 2 x cos x 1 0 (2 cos x 1)(cos x 1) 0 1 or 1 2 5 x 2 n, 2 n, 2 n 3 3 cos x Using Double-Angle Formulas to Analyze Graphs Use a double-angle formula to rewrite the equation. Then sketch the graph of the equation over the interval [0, 2 ] g ( x) 3 6sin 2 x g ( x) 3(1 2sin 2 x) g ( x) 3cos(2 x) Evaluating Functions Involving Double Angles Find sin(2x), cos(2x), tan(2x) 3 sin x , 0 x 5 2 4 5 3 4 24 sin(2 x) 2sin x cos x 2 5 5 25 Draw a picture to find : cos x 2 4 cos(2 x) 2 cos 2 x 1 2 1 5 7 16 2 1 25 25 24 sin(2 x) 25 24 tan(2 x) 7 cos(2 x) 7 25 Deriving a Triple-Angle Formula Derive the triple angle formula for cos(3x) cos 3x cos(2 x x) cos 2 x cos x sin 2 x sin x (2 cos 2 x 1) cos x (2sin x cos x)sin x 2 cos3 x cos x 2sin 2 x cos x 2 cos3 x cos x 2(1 cos 2 x) cos x 2 cos3 x cos x 2 cos x 2 cos3 x 4 cos3 x 3cos x Rewrite tan 4 x as a quotient of first powers of the cosines of multiple angles. tan 4 x tan 2 x 2 1 cos 2 x 1 cos 2 x 1 2 cos 2 x cos 2 2 x 1 2 cos 2 x cos 2 2 x 1 cos 4 x 1 2 cos 2 x 2 1 cos 4 x 1 2 cos 2 x 2 2 4 cos 2 x 1 cos 4 x 2 4 cos 2 x 1 cos 4 x 3 4 cos 2 x cos 4 x 3 4 cos 2 x cos 4 x 2 Using Half-Angle Formulas Find the exact value of cos 105 degrees. cos105 3 2 1 2 2 3 4 2 3 2 1 cos 210 2 Solving a Trigonometric Equation Find all the solutions of cos 2 x sin 2 x 2 1 cos x cos 2 x 2 2 2 cos x 1 cos x cos 2 x sin 2 2 cos 2 x cos x 1 0 (2 cos x 1)(cos x 1) 0 1 cos x or 1 2 5 x , , 3 3 Common Mistake: x cos 2 x 2 x 1 cos x 2 cos 2 2 2 2 2 cos 2 Rewrite Products as Sums Rewrite as a sum or a difference. sin 5 x cos 3 x 1 [sin(5 x 3 x) sin(5 x 3 x)] 2 1 [sin(8 x) sin(2 x)] 2 x on the interval from [0, 2 ] 2 Find the exact value of sin195+sin105 195 105 195 105 sin195 sin105 2sin cos 2 2 300 90 2sin cos 2 2 2sin150 cos 45 1 2 2 2 2 2 2 Solve: sin4x-sin2x=0 List all the values between 0 and Pi inclusive. sin 4 x sin 2 x 0 4x 2x 4x 2x 2 cos sin 2 2 6x 2x 2 cos sin 2 2 2 cos 3 x sin x 0 cos 3 x 0 3x x 2 6 x 0, sin x 0 n 3 x 0n n 5 , 6 2 , 6 , , 7 3 11 , , , 2 6 2 6 Example 11: Verify the Identity sin 6 x sin 4 x tan 5 x cos 6 x cos 4 x 6x 4x 6x 4x 2sin cos 2 2 6x 4x 6x 4x 2 cos cos 2 2 2sin 5 x cos x 2 cos 5 x cos x sin 5 x tan 5 x cos 5 x Homework #36 pp 415-416 1-6 all, 9-37 odd, 41-77 EOO, 95-100 all