Expert: Sherry Wallin - 2/10/2010

Question
1.2sin^2-sinx-1=0
2. 4sin^3x+2sin^2-2sinx-1=0

Answer
1.2sin^2-sinx-1=0
Just factor it: instead of sin x let y = sin x and now you have through substitution 2y^2-y-1=0 ->
(2y+1)(y-1)= 0 so 2y + 1 = 0 -> y = -1/2 and y = 1 which really means that the sinx = -1/2 and the
sinx = 1. The sinx is 1/2 at 30 degrees and it is -1/2 when you are in the 3rd or 4th quadrant, so the angle is either 210 degrees or 330 degrees (using reference angles with respect to the x-axis).

4sin^3x+2sin^2-2sinx-1=0
Us e the same trick let sinx = y so you have 4y^3 + 2y^2 - 2y - 1 = 0, factor by grouping:
2y^2(2y+1) - 1(2y+1)= 0 -> (2y^2 - 1) (2y + 1) = 0 so 2y^2 -1 = 0 -> 2y^2 = 1 -> y^2 = 1/2 ->
y = +-sqrt(1/2) -> sin x = +-sqrt(1/2). If you want to know what x is here take the arcsin of +-1/2
2y + 1 = 0 -> 2y = -1 -> y = -1/2 and the sinx = -1/2 the same as in the first problem.

Math Prof