Solve for θ if 0° ≤ θ < 360°. (Enter your answers as a comma-separated list.) sqrt 3 sin θ − cos θ = 1

Answer 1

Let's do this one, Madison!

√3 sinθ - cosθ = 1 . . . . . . . . . . . the √3 is a hint to square both sides of the equation

3 sin²θ - 2√3 sinθ cosθ + cos²θ = 1

2 sin²θ - 2√3 sinθ cosθ + (sin²θ + cos²θ) = 1 . . . . separating out one sin²θ for the next line

2 sin²θ - 2√3 sinθ cosθ + 1 = 1 . . . . . . . . . . since sin²θ + cos²θ for any angle θ

2 sin²θ - 2√3 sinθ cosθ = 0

2 sinθ (sinθ - √3 cosθ) = 0

=> 2 sinθ = 0 or (sinθ - √3 cosθ) = 0

sinθ = 0 or sinθ = √3 cosθ

θ = 0 or 180⁰ or tanθ = √3 or cosθ = 0 . . . . . . . if cosθ = 0, we may not divide by it!

θ = 0 or 180⁰ or θ = 60⁰ or 120^o or θ = 90⁰ or 270⁰

Solution set = {0, 60, 90, 120, 180, 270} all in degrees