Question

find the terminal point on the unit circle determined by $\frac{7pi}{4}$ radians. use exact values, not decimal approximations.

Explanation:

Step1: Recall the unit - circle formula

For a point $(x,y)$ on the unit circle determined by an angle $\theta$, $x = \cos\theta$ and $y=\sin\theta$. Here $\theta=\frac{7\pi}{4}$.

Step2: Find the x - coordinate

We know that $\cos\frac{7\pi}{4}=\cos(2\pi - \frac{\pi}{4})$. Since $\cos(2\pi-\alpha)=\cos\alpha$, then $\cos\frac{7\pi}{4}=\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}$.

Step3: Find the y - coordinate

We know that $\sin\frac{7\pi}{4}=\sin(2\pi - \frac{\pi}{4})$. Since $\sin(2\pi-\alpha)=-\sin\alpha$, then $\sin\frac{7\pi}{4}=-\sin\frac{\pi}{4}=-\frac{\sqrt{2}}{2}$.

Answer:

$(\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2})$