Question 1156517
start with 5 + 2 * cos(x) - 8 * sin^2(x) = 0
since sin^2(x) + cos^2(x) = 1, then sdin^2(x) = 1 - cos^2(x).
replace sin^2(x) with that to get:
5 + 2 * cos(x) - 8 * (1 - cos^2(x)) = 0
simplify to get:
5 + 2 * cos(x) - 8 + 8 * cos^2(x) = 0
order the equation in descending order of degree and combine like terms to get:
8 * cos^2(x) + 2 * cos(x) - 3 = 0
factor this quadratic equation to get:
cos(x) = -.75 or .5
assume these are both positive and solve for x in the first quadrant to get:
x = 41.40962211 degrees or x = 60 degrees.
cosine = -.75 is negative which means the angle has to be in the second and third quadrant.
x = 180 - 41.40962211 in the second quadrant = 138.5903779 degrees.
x = 180 + 41.40962211 in the third quadrant = 221.4096221 degrees.
cosine = .5 is positive which means the angle has to be in the first and fourth quadrant.
x = 60 degrees in the first quadrant.
x = 360 - 60 = 300 degrees in the fourth quadrant.
your angle can be 60, 138.5903779, 221.4096221, and 300 degrees.
this is confirmed by the graph of the equation shown below.


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