Question

what are the coordinates of the point that corresponds to $-\frac{5pi}{4}$ on the unit circle?
$(-\frac{sqrt{2}}{2}, -\frac{sqrt{2}}{2})$
$(\frac{sqrt{2}}{2}, -\frac{sqrt{2}}{2})$
$(-\frac{sqrt{2}}{2}, \frac{sqrt{2}}{2})$
$(\frac{sqrt{2}}{2}, \frac{sqrt{2}}{2})$

Explanation:

Step1: Recall unit - circle coordinate formula

For a point on the unit circle corresponding to an angle $\theta$, the coordinates are $(\cos\theta,\sin\theta)$. Here $\theta =-\frac{5\pi}{4}$.

Step2: Use the property of cosine and sine for negative angles

We know that $\cos(-\alpha)=\cos\alpha$ and $\sin(-\alpha)=-\sin\alpha$. So $\cos(-\frac{5\pi}{4})=\cos\frac{5\pi}{4}$ and $\sin(-\frac{5\pi}{4})=-\sin\frac{5\pi}{4}$.

Step3: Rewrite the angle

$\frac{5\pi}{4}=\pi+\frac{\pi}{4}$. Using the sum - of - angles formulas $\cos(A + B)=\cos A\cos B-\sin A\sin B$ and $\sin(A + B)=\sin A\cos B+\cos A\sin B$ with $A=\pi$ and $B = \frac{\pi}{4}$, we have $\cos(\pi+\frac{\pi}{4})=-\cos\frac{\pi}{4}=-\frac{\sqrt{2}}{2}$ and $\sin(\pi+\frac{\pi}{4})=-\sin\frac{\pi}{4}=-\frac{\sqrt{2}}{2}$. Then $\cos(-\frac{5\pi}{4})=-\frac{\sqrt{2}}{2}$ and $\sin(-\frac{5\pi}{4})=\frac{\sqrt{2}}{2}$.

Answer:

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