Question

Polar Coordinates to Rectangular Coordinates Find the rectangular coordinates for the point whose polar coordinates are given.$(\sqrt{3},-5\pi /3)$.

Short answer

The rectangular coordinates of the point $(\sqrt{3},-5\pi /3)$are$(\frac{\sqrt{3}}{2},\frac{3}{2})$

Step-by-step solution

Step 1.Given information.

The polar coordinates of the point are$(\sqrt{3},-5\pi /3)$ .

Step 2. Finding the value of x co-ordinate

Since the polar coordinates of the point are $(\sqrt{3},-5\pi /3)$we can say that$r=\sqrt{3}$and

$\begin{array}{l}\theta =\frac{-5\pi }{3}\\ \Rightarrow x=r\cos \theta \\ \Rightarrow x=\sqrt{3}\cos (-\frac{5\pi }{3})\\ \Rightarrow x=\sqrt{3}\cdot \left(\frac{1}{2}\right)\\ \Rightarrow x=\frac{\sqrt{3}}{2}\end{array}$

Step3. Finding the value of y co-ordinate

Now, to find the value of y,

$\begin{array}{l}\Rightarrow y=r\sin \theta \\ \Rightarrow y=\sqrt{3}\sin (-\frac{5\pi }{3})\\ \Rightarrow y=\sqrt{3}\cdot \left(\frac{\sqrt{3}}{2}\right)\\ \Rightarrow y=\frac{3}{2}\end{array}$