How do i find the solution to 2sin^2 x − sin x − 1 = 0

You might want to call y = sin x for a while, so that it looks less confusing. Then you have:

2y^2 - y - 1 = 0

You can either factorize or use the quadratic formula to find y. I'll factorize using an odd method:

(2y)^2 - (2y) - 2 = 0

(2y - 2)*(2y + 1) = 0

(y - 1)*(2y + 1) = 0

y1 = 1

y2 = -1/2

Since we said that y = sin x, we have:

sin x = 1 and sin x = -1/2

If sin x = 1, x = 90 degrees

If sin x = -1/2, there are two possibilities. Use mere inspection. We know that sin 30 = 1/2 but the answer is -1/2. Therefore x can be 180 + 30 = 210 degrees and x can be 360 - 30 = 330 degrees.

Your answers are:

x1 = 90 degrees
x2 = 210 degrees
x3 = 330 degrees

Please look at a chart. I hope this helps...