Solve the equation for 𝜃 if 0° ≤ 𝜃 < 360°. (Enter your answers as a comma-separated list.) 2 sin 𝜃 − 1 = csc 𝜃

Answer 1

I'll use T to stand for theta:

2sin(T) - 1 = csc(T)

Replace csc(T) with 1/sin(T)

2sin(T) - 1 = 1/sin(T)

Multiply both sides of equation by sin(T)

2sin²(T) - sin(T) = 1

Set equation equal to 0

2sin²(T) - sin(T) - 1 = 0

Factor

(2sin(T) + 1)(sin(T) - 1) = 0

Set each factor equal to zero and solve.

2sin(T) + 1 = 0 implies that sin(T) = -1/2. Using the unit circle, we know that T = 210 and 330 degrees

sin(T) - 1 = 0 implies that sin(T) = 1. Again from the unit circle, we knot T = 90 degrees

Hence, all three solutions are T = 90, 210, and 330 degrees.