Solving a Trigonometric Equation in Quadratic Form

We begin by using substitution and replacing cos $\theta$ with $x.$ It is not necessary to use substitution, but it may make the problem easier to solve visually. Let $\cos \theta =x.$ We have

The equation cannot be factored, so we will use the quadratic formula $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}.$

Replace $x$ with $\cos \theta ,$ and solve.

Note that only the + sign is used. This is because we get an error when we solve $\theta ={\cos }^{-1}\left(\frac{-3-\sqrt{13}}{2}\right)$ on a calculator, since the domain of the inverse cosine function is $\left[-1,1\right].$ However, there is a second solution:

This terminal side of the angle lies in quadrant I. Since cosine is also positive in quadrant IV, the second solution is