Solution

Find all solutions to the following equation

$$\sin \theta = \cos \theta$$

There are two trigonometric functions involved in the above equation. Ideally, we would prefer to only have one.

One way we could proceed is to cleverly notice that if we divide both sides by $\cos \theta$, the cosine on the right side dissappears, and the left side gets turned into a tangent, as shown

$$\begin{array}{rcl} \frac{\sin \theta}{\cos \theta} &=& \frac{\cos \theta}{\cos \theta}\\\\ \tan \theta &=& 1 \end{array}$$

Where this occurs within one full counter-clockwise rotation on the unit circle (i.e., where $0 \le \theta \lt 2\pi$) should be familiar -- namely, $\theta = \pi/4$ and $\theta = 5\pi/4$.

Then, accounting for angles co-terminal to these two values of $\theta$, we arrive at a solution of

$$\theta = \frac{\pi}{4} \pm 2\pi n \,\, , \,\, \frac{5\pi}{4} \pm 2\pi n \quad \textrm{where } n \textrm{ is an integer}$$


Another way to proceed is to remember that sines represent $y$-coordinates of points on the unit circle, while cosines represent $x$-coordinates of the same.

Thus, if we seek an angle $\theta$ where $\sin \theta = \cos \theta$, we seek a point $(x,y)$ on the unit circle where $y=x$.

Equivalently, we seek the points of intersection between the unit circle and the line $y=x$.

Drawing the line $y=x$ on top of the unit circle and remembering that it forms a $45^{\circ}$ angle (i.e., $\pi/4$ radians) with the positive $x$-axis, we can quickly deduce that the only solutions to our equation in $[0,2\pi)$ are again $\theta = \pi/4$ and $\theta = 5\pi/4$.

After addressing the issue of co-terminal angles again, we end up with the exact same solution as found above,

$$\theta = \frac{\pi}{4} \pm 2\pi n \,\, , \,\, \frac{5\pi}{4} \pm 2\pi n \quad \textrm{where } n \textrm{ is an integer}$$