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Advanced Algebra Trig Lesson Plans Section 7.5 – Multiple-Angle and Product-to-Sum Formulas Enduring Understandings: The student shall: 1. know and use double-angle formulas 2. be able to use power-reducing formulas 3. be able to use half-angle formulas to rewrite and evaluate functions 4. be able to use product-to-sum formulas to rewrite and evaluate functions Standards: 17 Trigonometry Simplifies trigonometric expressions and solve trigonometric equations using trigonometric identities Essential Questions: 1. How do we deal with multiple angles? Activities: Double-Angle Formulas – MEMORIZE THE SINE AND COSINE DOUBLE ANGLE FORMULAS and be able to use the rest. Double Angle Formulas: Let both angles in the Sum and Difference Formulas equal each other. sin 2u = 2 sin u cos u tan 2u 2 tan u 1 tan 2 u cos 2u = cos2 u – sin2 u = 2 cos2 u – 1 = 1 – 2 sin2 u Power-Reducing Formulas: Solve the above for the squared function. sin 2 u 1 cos 2u 2 cos 2 u 1 cos 2u 2 tan 2 1 cos 2u 1 cos 2u Half-Angle Formulas: Substitute v = u/2 in the above equations and then take square root. u 1 cos u sin u u 1 cos u tan 2 sin u 1 cos u 2 2 u 1 cos u cos 2 2 NOTE: the signs of sin(u/2) and cos(u/2) depend on the quadrant in which u/2 lies. sin Product-to-Sum Formulas: Easily verified from Sum and Difference Formulas in last section. 1 1 sin u sin v cosu v cosu v cos u cos v cosu v cosu v 2 2 1 1 sin u cos v sin u v sin u v cos u sin v sin u v sin u v 2 2 Sum-to-Product Formulas: Let x = u + v, and y = u – v, solve for u and v, and substitute into the product to sum formula. x y x y sin x sin y 2sin cos 2 2 x y x y sin x sin y 2cos sin 2 2 x y x y cos x cos y 2cos cos 2 2 x y x y cos x cos y 2sin sin 2 2 Do: Day One: 4, 6, 10, 12, 20, 24, 32, 36, 46, 50, 60, 64, Do: Day Two: 78, 80, 88, 92, 96, 114, 118, 120 Assessments: Hw pg 570 - 573, # 1 – 73 by 4’s (19) Hw pg 570 - 573, # 77 – 121 by 4’s (12)