Well, first of all, the equation in your question isn't clear. I decided that you intended to write the equation
-20 cos²(x) - 3 cos(x) = -2
If this is not the correct equation, then you should stop reading right here. Everything else from here on is nonsense, and you should report my answer.
= = = = =
If that IS the correct equation, then ...
Multiply each side by -1 . . . 20 cos²(x) + 3 cos(x) = 2
Subtract 2 from each side . . . 20 cos²(x) + 3 cos(x) - 2 = 0
This is a quadratic equation in 'cos(x)' . I messed around for a while trying to factor it, and then I gave up and used the Quadratic Formula.
Like any other quadratic equation, this one has two roots:
(1). cos(x) = -119/40 or -0.475
(2). cos(x) = 7/40 or 0.175
For the first one, the angle 'x' is 118.36° or 241.64°
For the second one, the angle 'x' is 79.92° or 280.08°
So the original equation ... the one with all the cosines in it ... actually has four (4) solutions.
