1. 5.5 Solving Trigonometric Equations Example 1 A) Is a solution to ? B) Is a solution to cos x = sin 2x ?

2. Solving Trigonometric Equations - Overview

3. Trigonometric Equations with a Single Trig Function For equations with a single trig function, isolate the trig function on one side. Solve for the variable by identifying the appropriate angles. Be prepared to express your answer in radian measure.

4. Example 2 Find all solutions for

5. Example 2 - Solution where n is any integer

6. Example 3 Solve the equation on the interval [0º, 360º) sin x = x = 30º, 150º

7. Other Strategies for Solving Solving Trig Equations Put the equation in terms of one trig function (if possible). Solve for the trig function (using algebra – addition, subtraction, multiplication, division, factoring). Solve for the variable (using inverse trig functions, reference angles). Use a fundamental identity to end up with a single trig function.

8. Example 4 To solve an equation containing a single trig function: Solve: 3sinx – 2 = 5sinx - 1 * Isolate the function on one side of the equation. * Solve for the variable. Solution: 3sinx - 5sinx = -1 +2 -2sinx = 1 sinx = -1/2 (Remember: x are the angles whose sine is -1/2)

9. Example 5 Solve the equation on the interval [0, 2π) 2 cos x − 1 = 0 2 cos x = 1 cos x = x =

10. Example 6 -Trigonometric Equations Quadratic in Form. Ans. π/6, π/2, 5π/6 Try to solve by factoring It factors in the same manner as = (2x -1)(x – 1) Solution: (2sinx – 1)(sinx -1) = 0 2sinx – 1 = 0 2sinx = 1 sinx = ½ Therefore x = π/6, 5π/6 sinx – 1 = 0 sinx = 1 x = π/2

12. Example 8: Solve an Equation with a Multiple Angle.

13. Ans. x = Example 9 - Multiple Angle

14. Ans. 0, π Move all terms to one side, then factor out a common trig function. Example 10

15. Ans. π/3, 5π/3 The equation contains more than one trig function; there is no common trig function. Try using an identity. Example 11

16. Example 12 Solve the equation: cos2x + 3sinx – 2 = 0, 0 ≤ x ≤ 2π Ans. π/6, π/2, and 5π/6

17. Example 13 Solve the equation: sinx cosx= -1/2, 0 ≤ x ≤ 2π Ans. 3π/4, 7π/4

18. Example 14 - using a calculator to solve Solve the equation correct to four decimal places, 0 ≤ x ≤ 2π a. tan x = 3.1044 b. sin x = -0.2315 Ans. a. 1.2592, 4.4008 b. 3.3752, 6.0496 Use a calculator to find the reference angle, then use your knowledge of signs of trigonometric functions to find x in the required interval.

19. Ans. 2.3423, 3.9409 The equation is in quadratic form, but does not factor. Use the quadratic formula to solve for the trig function of x, then use a calculator and the Example 15