Givens
sin(2x) = 2 * sin(x) * cos(x)
Equation
sin(x)*sin2(X) = 2sin(x) * 2sin(x)*cos(x) = 1
4*sin^2(x)* cos(x) = 1
sin^2(x) = 1 - cos^2(x)
4(1 - cos^2(x) ) * cos(x) = 1 Divide by 4
(1 - cos(x) ) *cos(x) = 1/4
cos(x) - cos^3(x) = 1/4
Let y = cos(x)
y - y^3 = 1/4
This is a cubic. I have to use a calculator's built in formula to solve it.
a = -1
b = 0
c = 1
d = - 1/4
Solution
y1 = - 1.107. Extraneous cos(x) can't be less than 1.
y2 = 0.8376
y3 = 0.2696
but cos(x) = y
cos-1(0.8376) = 33.11 degrees.
cos-1(0.2696) = 74.36 degrees.
The cosine is also plus in quad 4.
x = 360 - 33.11 = 326.89
x = 360 - 74.36 = 285.64
The graph of the cubic is below. It pretty much confirms what I found.
