Solution

Find all solutions to the following equation

$$3\cos^2 \theta - \cos 2\theta = 1$$

While there is only one trigonometric function involved, it is being applied to different angles (i.e., $\theta$ in one case, $2\theta$ in the other). We would prefer only a single angle measure to be present, as this generally makes solving the equation easier.

Consequently, we use one of the double angle identities to replace the $\cos 2\theta$ with something more suitable. Recall, we have three relevant double angle identities for $\cos 2\theta$ that could be used to this end:

$$\begin{array}{c} \cos 2\theta &=& \cos^2 \theta - \sin^2 \theta\\ &=& 2\cos^2 \theta - 1\\ &=& 1 - 2\sin^2 \theta \end{array}$$

We prefer not to complicate the equation by introducing a second trigonometric function (i.e., $\sin \theta$). As such, the second identity in the list above is the most appropriate to use. Making the substitution and simplifying the resulting equation, we have

$$\begin{array}{rcl} 3\cos^2 \theta - \cos 2\theta &=& 1\\ 3\cos^2 \theta - (2\cos^2 \theta - 1) &=& 1\\ \cos^2 \theta + 1 &=& 1\\ \cos^2 \theta &=& 0\\ \cos \theta &=& 0 \end{array}$$

Now we can appeal to the unit circle to determine where the cosine (i.e., an $x$-coordinate) is zero:

As seen above, in the interval $[0,2\pi)$ there are only two solutions, $\theta = \pi/2$ and $\theta = 3\pi/2$.

Accounting for angles co-terminal to these, we have the set of all solutions given by:

$$\theta = \frac{\pi}{2} \pm 2\pi n \, \, , \, \, \frac{3\pi}{2} \pm 2\pi n \quad \textrm{where } n \textrm{ is an integer}$$

We could stop there, however...

We can write things more succinctly if we take advantage of some symmetry present in this solution set. Notice that one can always get from one solution to another by adding $\pi$: $$\begin{array}{rcll} \frac{\pi}{2} + \pi &=& \frac{3\pi}{2}\\\\ \quad \frac{3\pi}{2} + \pi &=& \frac{5\pi}{2} \quad \scriptsize{\textrm{(a coterminal to } \frac{\pi}{2} \textrm{)}}\\\\ \quad \frac{5\pi}{2} + \pi &=& \frac{7\pi}{2} \quad \scriptsize{\textrm{(a coterminal to } \frac{3\pi}{2} \textrm{)}}\\\\ & \vdots & \end{array}$$

Indeed, the difference between any two solutions is always an integer multiple of $\pi$.

Thus, the following also describes our solution set:

$$\theta = \frac{\pi}{2} \pm \pi n \quad \textrm{where } n \textrm{ is an integer}$$