since sin^2(x) is equal to 1 - cos^2(x), then you equation can be converted to:
5 - 5cos^2(x) - 5cos(x) = -3
multiply both sides of the equation by -1 and you get:
-5 + 5cos^2(x) + 5cos(x) = 3
subtract 3 from both sides of the equation and re-arrange the terms in descending order of exponential degree and you get:
5cos^2(x) + 5cos(x) - 8 = 0
this is a quadratic equation that you can solve using the quadratic equation.
you will get:
cos(x) = .8601470509
cos(x) = -1.860147051
solve for x by taking the arc-cosine of those values and you get:
x = 30.66690224 degrees in the first quadrant.
x = invalid value for arc-cosine (-1.86...) because the maximum and minimum value of cos(x) is between -1 and 1.
so you have one solution.
x = 30.66690224 degrees.
that's in the first quadrant.
cosine is also positive in the fourth quadrant.
value of x in the fourth quadrant is 360 - 30.66690224 degrees = 329.3330978.
in the interval between 0 degrees and 360 degrees you have two solutions for x that will satisfy the equation.
they are:
x = 30.66690224 degrees
x = 329.3330978 degrees
here's a graph of the equations of:
y = 5sin^2(x) - 5cos(x)
y = -3
the intersections are where both equations have the same value of y for the same values of x.
