Question

2 - 48. sketch each of the following angles in a unit circle. then state the coordinates of the corresponding point on the unit circle. homework help a. $\theta=\frac{5pi}{3}$ b. $\theta =-\frac{5pi}{4}$ c. $\theta=\frac{8pi}{3}$

Explanation:

Step1: Recall the unit - circle coordinate formula

For an angle $\theta$ in standard position, the coordinates of the corresponding point on the unit circle are given by $(x,y)=(\cos\theta,\sin\theta)$.

Step2: Calculate for $\theta=\frac{5\pi}{3}$

We know that $\cos\frac{5\pi}{3}=\cos(2\pi - \frac{\pi}{3})=\cos\frac{\pi}{3}=\frac{1}{2}$ and $\sin\frac{5\pi}{3}=\sin(2\pi-\frac{\pi}{3})=-\sin\frac{\pi}{3}=-\frac{\sqrt{3}}{2}$. So the coordinates are $(\frac{1}{2},-\frac{\sqrt{3}}{2})$.

Step3: Calculate for $\theta =-\frac{5\pi}{4}$

$\cos(-\frac{5\pi}{4})=\cos\frac{5\pi}{4}=\cos(\pi+\frac{\pi}{4})=-\cos\frac{\pi}{4}=-\frac{\sqrt{2}}{2}$ and $\sin(-\frac{5\pi}{4})=-\sin\frac{5\pi}{4}=-\sin(\pi + \frac{\pi}{4})=\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}$. So the coordinates are $(-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2})$.

Step4: Calculate for $\theta=\frac{8\pi}{3}$

$\frac{8\pi}{3}=2\pi+\frac{2\pi}{3}$. Then $\cos\frac{8\pi}{3}=\cos(2\pi+\frac{2\pi}{3})=\cos\frac{2\pi}{3}=-\frac{1}{2}$ and $\sin\frac{8\pi}{3}=\sin(2\pi+\frac{2\pi}{3})=\sin\frac{2\pi}{3}=\frac{\sqrt{3}}{2}$. So the coordinates are $(-\frac{1}{2},\frac{\sqrt{3}}{2})$.

Answer:

a. Coordinates: $(\frac{1}{2},-\frac{\sqrt{3}}{2})$
b. Coordinates: $(-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2})$
c. Coordinates: $(-\frac{1}{2},\frac{\sqrt{3}}{2})$