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Double and Half Angle Formulas ** More trigonometric formulas - based on the addition formulas from previous section (Gotta Love Them). Recall: sin x y cos x y tan x y sin 2 cos 2 tan 2 7.5_doublehalfangleBLANK Page 1 Double Formulas Example: If sin 3 , , find the exact value of sin 2 and cos 2 5 2 Example: Find an expression for the triple angle formula cos 3x Example: Find the exact value of the following expression: 7.5_doublehalfangleBLANK Page 2 in terms of cosine only. 5 sin 2 cos 1 13 Double Angle Formulas As a consequence of the double angle formulas from cos 2 , we can construct the following useful identities. sin 2 cos 2 tan 2 Example: Simplify the following using the double angle formulas until the powers of sine and cosine are not greater than 1: sin 2 2 x cos 2 2 x 7.5_doublehalfangleBLANK Page 3 Half Angle Formulas As a consequence of the double angle formulas on the previous page, we can construct the half-angle formulas: 1 1 cos u 1 cos u 2 2 1 1 sin u 1 cos u 2 2 sin u 1 1 cos u tan u sin u 1 cos u 2 The +/- depends on the quadrant in which 1 u lies. 2 Example: Find the exact value of (a) cos 15 (b) Example: Find the exact value of the expression: 1 3 cos 2 sin 1 5 2 7.5_doublehalfangleBLANK Page 4 7 csc 8 Double and Half Angle Formulas Establish the identity: cot tan cos 2 cot tan Example: If x 8 tan , express 7.5_doublehalfangleBLANK Page 5 sin 2 as a function of x.