1. Trigonometric Equations

2. Equations Involving a Single Trigonometric Function To solve an equation containing a single trigonometric function: • Isolate the function on one side of the equation. • Solve for the variable.

3. x Trigonometric Equations y y = cos x 1 y = 0.5 x –4  –2  2  4  –1 cos x = 0.5 has infinitely many solutions for –< x <  y y = cos x 1 0.5 2  cos x = 0.5 has two solutions for 0 < x < 2 –1

4. This is the given equation. 3 sin x- 2 = 5 sin x- 1 Subtract 5 sin x from both sides. 3 sin x- 5 sin x- 2 = 5 sin x- 5 sin x – 1 Simplify. -2 sin x- 2 =-1 Add 2 to both sides. -2 sin x= 1 Divide both sides by -2 and solve for sin x. sin x= -1/2 Text Example Solve the equation: 3 sin x- 2 = 5 sin x- 1. Solution The equation contains a single trigonometric function, sin x. Step 1Isolate the function on one side of the equation. We can solve for sin x by collecting all terms with sin x on the left side, and all the constant terms on the right side.

5. Solution The given equation is in quadratic form 2t2+t- 1 = 0 with t= cos x. Let us attempt to solve the equation using factoring. This is the given equation. 2 cos2x+ cos x- 1 = 0 Factor. Notice that 2t2 + t – 1 factors as (2t – 1)(2t + 1). (2 cos x- 1)(cos x+ 1) = 0 Set each factor equal to 0. 2 cos x- 1= 0 or cos x+ 1 = 0 Solve for cos x. 2 cos x= 1 cos x= -1 Text Example Solve the equation: 2 cos2 x+ cos x- 1 = 0, 0 £x< 2p. cos x= 1/2 x=px= 2pppx=p The solutions in the interval [0, 2p) are p/3, p, and 5p/3.

6. Example • Solve the following equation: Solution:

7. Example • Solve the equation on the interval [0,2) Solution:

8. Example • Solve the equation on the interval [0,2) Solution:

9. Example • Solve the equation on the interval [0,2) Solution:

10. Trigonometric Equations