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Review – Trigonometry • • • Trigonometric Identities Trigonometric Equations Double Angle Identities 2 2 cos θ + sin θ = 1 2 2 1 + tan θ = sec θ 2 2 cot + 1 = csc θ sin2θ = 2sinθcosθ 2 2 cos 2θ = cos θ - sin θ 2 = 1 – 2sin θ 2 = 2cos θ - 1 TRIGONOMETRIC IDENTIES Prove the following trigonometric identities. 1. cos x + cot x sin x = 2 sin x cot x 2. 2 cot x = 2 cos 2 x cot x + tan x 3. 1 + sin x 1 + sin x = 1 − sin x cos x 4. 1 cos x − = 2 csc 2 x − 1 1 − cos x 1 + cos x 5. sec x − tan x = 1 − sin x cos x TRIGONOMETRIC EQUATIONS Solve the following equations if 0 < x < 2π. 1. 4cos2x – 3 = 0 2. 2sin3x = sinx 3. 2sin2x = 1 – cosx 4. 8sin2xcosx – 2cosx – 4sin2x + 1 = 0 Hint: use factoring by groupings DOUBLE ANGLE IDENTITIES 1. If sin x = 2/3, x in QII, find sin 2x and cos 2x. 2. If cos x = − 5 , x in QIII, find sin 2x and cos 2x. 2 CIRCULAR FUNCTIONS x2 + y2 = r2 cos θ = x r sin θ = x r r y θ x 1. If sin θ = 1 , θ in QII, find the remaining trigonometric functions. 3 2. If tan θ = 5 , θ in QIII, find the remaining trigonometric functions. 7 3. If sec = 4 , θ in QI, find the remaining trigonometric functions. 3 tan θ = y x