Dayaan M. answered • 08/26/26
Experienced Math and Computer Science Tutor - Helping Students Excel
In order to solve an equation like this, the first thing to notice is that it has both sine and cosine in it, and that is the real problem. We cannot do much while there are two different trig functions floating around, so step one is always to rewrite it so everything is in terms of just one of them.
The Pythagorean identity sin^2(theta) + cos^2(theta) = 1 lets us do exactly that. Rearranged, it says sin^2(theta) = 1 - cos^2(theta), and since our equation has sin^2 rather than a lone sin, we can substitute directly:
6(1 - cos^2(theta)) + 7cos(theta) - 8 = 0
Distributing the 6 and combining the constants:
6 - 6cos^2(theta) + 7cos(theta) - 8 = 0
-6cos^2(theta) + 7cos(theta) - 2 = 0
I like to multiply through by -1 here so the leading term is positive, since it makes the factoring easier to see:
6cos^2(theta) - 7cos(theta) + 2 = 0
If you notice, this is just a quadratic. If it helps, imagine cos(theta) is x and read it as 6x^2 - 7x + 2 = 0. Factoring gives:
(3cos(theta) - 2)(2cos(theta) - 1) = 0
So either cos(theta) = 2/3 or cos(theta) = 1/2.
Remember, our goal is every angle between 0 and 360 degrees, so each of these gives us two answers, not one. Cosine is positive in quadrant I and quadrant IV, so for each value we take the reference angle and then subtract it from 360 to get the second one.
For cos(theta) = 1/2, that is one of the special angles, so theta = 60 degrees and theta = 360 - 60 = 300 degrees.
For cos(theta) = 2/3, the calculator gives an inverse cosine of about 48.2 degrees, so theta = 48.2 degrees and theta = 360 - 48.2 = 311.8 degrees.
So, our final answer is theta = 48.2, 60, 300 and 311.8 degrees.
