Coterminal and Reference Angles Exercise 3

Answer: \(\dfrac{\pi}{4}\)

Solution Method: First, to draw the angle in standard position we need to place its vertex on the origin, and its initial side on the positive x-axis. Then we just determine where the terminal side needs to be. The \(\frac{7\pi}{4}\) angle is positive so it's measured counterclockwise from the initial side. Each quadrant is \(\frac{\pi}{2}\) so \(\frac{7\pi}{4}\) lies between \(\frac{3\pi}{2}\) and \(2\pi\) in Q4.  

Now to find the reference angle. Recall, the reference angle is the smallest positive acute angle formed by the x-axis and the terminal side of the angle in standard position. The \(\frac{7\pi}{4}\) is in Q4 so the reference angle is drawn between the terminal side and the positive x-axis here. We see then that the reference angle is the difference of \(2\pi\) and \(\frac{7\pi}{4}\) 
\begin{align*}
    2\pi - \frac{7\pi}{4} &= \frac{8\pi}{4}-\frac{7\pi}{4} \\
    &= \frac{\pi}{4}
\end{align*}