Question

find the coordinates of the point p on the circumference of the circle. (hint: add x - and y - axes, assuming that the angle is in standard position.) the coordinates of p are (type an ordered pair. simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Recall polar - to - rectangular conversion

For a point $(r,\theta)$ in polar coordinates, the rectangular coordinates $(x,y)$ are given by $x = r\cos\theta$ and $y = r\sin\theta$. Here, $r = 8$ and $\theta=150^{\circ}$.

Step2: Calculate the x - coordinate

First, convert $\theta = 150^{\circ}$ to radians: $150^{\circ}=150\times\frac{\pi}{180}=\frac{5\pi}{6}$. Then, $x = r\cos\theta=8\cos150^{\circ}=8\cos\frac{5\pi}{6}$. Since $\cos\frac{5\pi}{6}=-\frac{\sqrt{3}}{2}$, $x = 8\times(-\frac{\sqrt{3}}{2})=- 4\sqrt{3}$.

Step3: Calculate the y - coordinate

$y = r\sin\theta=8\sin150^{\circ}=8\sin\frac{5\pi}{6}$. Since $\sin\frac{5\pi}{6}=\frac{1}{2}$, $y = 8\times\frac{1}{2}=4$.

Answer:

$(-4\sqrt{3},4)$