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6. AccPreCalc Lesson 27 Essential Question: How are trigonometric equations solved? Standards: Prove and apply trigonometric identities MCC9 ‐ 12.F.TF.9 (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

7. Solving Trig Equations Solve: 2sin( Ѳ) -1 = 0 2Sin( Ѳ ) = 1 Sin( Ѳ ) = ½ Ѳ = ∏/6, 5 ∏/6, 13 ∏/6, …. Restrict all solution from 0 to 2 ∏

8. Prove the following 1 + cos(2x) = cot(x) Sin(2x) 1 + cos 2 (x) – sin 2 (x) cos(x) / sin(x) 2sin(x) cos(x) cos 2 (x) + cos 2 (x) 2sin(x) cos(x) 2cos 2 (x) 2sin(x) cos(x) Cos(x) / sin(x) QED

9. Solve the equation Sin(x) + sqrt(2) = -Sin(X) 2sin(x) + sqrt(2) = 0 2sin(x) = -sqrt (2) Sin(x) = -sqrt(2) / 2 X = 5∏/4, 7∏/4 Check by plugging back into the equation

10. Solve the equation 3tan 2 (x) -1 = 0 Tan 2 (x) = 1/3 Tan(x) = +/_ 1/sqrt(3) rationalize your answer Tan(x) = +/- sqrt(3) / 3 X = ∏/6, 5∏/6, 7 ∏/6, 11 ∏/6

11. Solve for x Cot(x) cos 2 (x) = 2cot(x) Cot(x) cos 2 (x) - 2 cot(x) = 0 Cot(x) [cos 2 (x) -2] = 0 Set each factor equal to zero Cot(x) = 0 cos 2 (x) -2 = 0 X = ∏/ 2, cos 2 (x) = 2 3 ∏ /2 COS(X) = +/- SQRT(2) No solutions

12. Solving multiple angle equations Sin(2x) = 1 2x = ∏/ 22x = 5 ∏/ 2 X = ∏/4x = 5 ∏/4 2x = 9 ∏/2 X = 9 ∏/4 ….. This is bigger than 2 ∏ so this is not one of our solutions.

13. Multiple angle equation Cos(3x) = - sqrt(3) / 2 3x = 5 ∏/6 3x = 7 ∏/63x = 31 ∏/6 3x = 17 ∏/63x = 19 ∏/6 3x = 29 ∏/6

14. Solve 2sin 2 (2x) = 1 on 0 to ∏ sin 2 (2x)= ½ Sin(2x) = +/- sqrt(2) / 2 2x = ∏/42x = 3 ∏ /4 2x = 5 ∏ /4 2x = 7 ∏ /4 2x = 9 ∏ /4