7.5 SOLVING TRIGONOMETRIC EQUATIONS

When we solve a trigonometric equation, there will be infinite solutions because of the periodic nature of the function (repeating itself). Therefore, we often restrict answers. Be careful when solving equations! If asked to restrict to principal values, they are:

Principal Values:

Sine: -90°≤ x ≤90°

Cosine: 0° ≤ x ≤180°

Tangent: -90° ≤ x ≤90°

Practice:

1) Solve for principal values:

sinθcosθ – ½ cosθ = 0

2) 2sin2 θ+ sinθ = 0 for 0≤θ≤2π

3) Solve cos2x – cos x + 1 = sin2x for 0≤x≤2π

• Sin2x = -sinx for for °0≤ x ≤360°

Sometimes, we are asked to find ALL solutions. If that is the case, you can write the solution as x+360k (for sine cosine) or x + 180k (for tan)

4) Solve 2 sec2x – tan4x = -1 for ALL real values of x

5) Solve 2sinθ + 1 > 0 for 0≤x≤2π