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Double Angle Formulas T, 11.0: Students demonstrate an understanding of half-angle and doubleangle formulas for sines and cosines and can use those formulas to prove and/or simplify other trigonometric identities. Double Angle Formulas Objectives • Know the double angle identities • Verify more complex trigonometric identities using the basic trigonometric identities, Pythagorean identities, co-function identities, odd/even identities, sum & difference identities, double angle identities, halfangle identities and justify each step in the verification process. Key Words • Degrees • Radians • Double-Angle Identities Quick Check Solve Solve Double Angle Identities Using the Double Angle Formula 2 and has its terminal side 5 in the first quadrant, find the exact value of each function. If sin = a. sin 2 To use the double-angle identity for sin 2, we must first find cos . sin2 + cos2 = 1 2 2 + cos2 = 1 5 21 cos2 = 25 21 cos = 5 sin = 2 5 Then find sin 2. sin 2 = 2 sin cos 2 21 2 21 = 2(5)( 5 ) sin = 5, cos = 5 4 21 = 25 b. cos 2 Since we know the values of cos and sin , we can use any of the double-angle identities for cosine. cos 2 = 2 cos2 - 1 2 = 2 21 - 1 5 17 = 25 21 cos = 5 Using the Double Angle Formula 2 If sin = and has its terminal side 5 in the first quadrant, find the exact value of each function. c. tan 2 We must find tan to use the double-angle identity for tan 2. sin tan = cos 2 5 2 21 = sin = 5, cos = 5 21 5 2 2 21 = or 21 21 Then find tan 2. 2 tan tan 2 = 1 - tan2 2 2 21 2 21 21 = tan = 21 2 1 2 21 21 4 21 21 4 21 = 17 or 17 21 d. sin 4 Since 4 = 2(2), use a double-angle identity for sine again. sin 4 = sin 2(2) = 2 sin (2) cos (2) 4 21 17 = 2( 25 )(25) 136 21 = 625 Double-angle identity 4 21 17 sin (2) = 25 , cos (2) = 25 You Try 2 If sin 𝜃 = and 𝜃 has its 3 terminal side in the first quadrant, find the exact value of each function. a) sin 2𝜃 b) cos 2𝜃 c) tan 2𝜃 d) cos 4𝜃 You Try 2 If sin 𝜃 = and 𝜃 has its 3 terminal side in the first quadrant, find the exact value of each function. a) sin 2𝜃 b) cos 2𝜃 c) tan 2𝜃 d) cos 4𝜃 Verify Trigonometric Identity -1 - cos 2 Verify that = -cot is an identity. sin 2 -1 - cos 2 ? -cot sin 2 -1 - (2 cos2 - 1) ? -cot 2 sin cos -2 cos2 ? -cot 2 sin cos cos ? -cot sin -cot = -cot Double-angle identities for cosine and sine You Try Conclusions Summary • Verify that Sin2x cosx – cos2x sinx = sinx is an identity. Assignment • Double Angle Formulas – Page 453 – #(8-12,21-23,28-30)