4 * sin^2(x) - 8 * cos(2x) = 0
x represents the angle theta.
add 8 * cos(2x) to both sides of this equation and you get:
4 * sin^2(x) = 8 * cos(2x)
cos(2x) is equal to cos^2(x) - sin^2(x)
cos^2(x) is equal to 1 - sin^2(x)
cos(2x) is therefore equal to 1 - sin^2(x) - sin^2(x).
this simplifies to cos(2x) = 1 - 2 * sin^2(x)
your equation becomes:
4 * sin^2(x) = 8 * (1 - 2 * sin^2(x))
simplify this to get 4 * sin^2(x) = 8 - 16 * sin^2(x)
add 16 * sin^2(x) to both sides of this equation to get:
20 * sin^2(x) = 8
divide both sides of this equation by 20 to get sin^2(x) = 8/20.
take the square root of both sides of this equation to get sin(x) = sqrt(8/20)
use your calculator to find the arcsin of sqrt(8/20).
you will find that the angle is equal to 39.23152048 degrees.
the equivalent angle in the second quadrant is 180 minus that.
the equivalent angle in the third quadrant is 180 plus that.
the equivalent angle angle in the fourth quadrant is 360 minus that.
the equivalent angles in each quadrant are therefore.
quadrant 1 = 39.23152048
quadrant 2 = 140.7684795
quadrant 3 = 218,2315295
quadrant 4 = 320.7684795
you want to check that 4 * sin^2(x) = 8 * cos(2x) = 0 for each of these angles.
quadrant 1 = 0
quadrant 2 = 0
quadrant 3 = 0
quadrant 4 = 0
the formula checks out ok in all 4 quadrants, therefore your solution is the angles in each of the quadrant.
this is confirmed by the following graph of y = 4 * sin^2(x) and y = 8 * cos(2x).
the intersection of these 2 equations is your solution and it confirms that the angle in each of those quadrants provides the correct solution.
here's the graph.
