Question

22 (4 points) change (5, $\frac{5pi}{6}$) from polar to rectangular coordinates. list exact values. (x,y)=( , ) hide hint

Explanation:

Step1: Recall conversion formulas

The conversion formulas from polar coordinates $(r,\theta)$ to rectangular coordinates $(x,y)$ are $x = r\cos\theta$ and $y = r\sin\theta$. Here $r = 5$ and $\theta=\frac{5\pi}{6}$.

Step2: Calculate the x - coordinate

$x=r\cos\theta=5\cos\frac{5\pi}{6}$. Since $\cos\frac{5\pi}{6}=-\frac{\sqrt{3}}{2}$, then $x = 5\times(-\frac{\sqrt{3}}{2})=-\frac{5\sqrt{3}}{2}$.

Step3: Calculate the y - coordinate

$y = r\sin\theta=5\sin\frac{5\pi}{6}$. Since $\sin\frac{5\pi}{6}=\frac{1}{2}$, then $y = 5\times\frac{1}{2}=\frac{5}{2}$.

Answer:

$(-\frac{5\sqrt{3}}{2},\frac{5}{2})$