Solve the equation for x if 0 ≤ x < 2π. Give your answer in radians using exact values only. (Enter your answers as a comma-separated list.) 2 sin x − 3 cot x − 3 csc x = 0

Answer 1

Hi Madison:

2sinx - 3 cot x - 3 cosec x = 0 . . . . with cases like this, it's usually good to reduce all trig ratios to sin and cos 2 sinx - 3cosx/sinx - 3/sinx = 0

(2sin²x - 3cosx - 3) / sinx = 0 . . . common denominator is sinx

=> (2sin²x - 3cosx - 3) = 0

2(1 - cos²x) - 3 cosx - 3 = 0 . . . . replacing sin²x by (1 - cos²x) to get a quadratic in cosx

2cos²x + 3 cosx + 3 -2 = 0

2cos²x + 3 cosx + 1 = 0

(2cosx + 1)(cosx +1) = 0

=> 2cosx + 1 = 0 or cosx + 1 = 0

cosx = -1/2 or cosx = -1

(x = 2π/3 or 5π/6) or (x = π/2 or 3π/2)

Solution: x = π/2, 2π/3, 5π/6, 3π/2