Question

Polar Coordinates to Rectangular Coordinates Find the rectangular coordinates for the point whose polar coordinates are given.$(6,2\pi /3)$.

Short answer

The rectangular coordinates of the point $(6,\frac{2\pi }{3})$are$(-3,3\sqrt{3})$

Step-by-step solution

Step 1.Given information.

The polar coordinates of the point are$(6,\frac{2\pi }{3})$ .

Step 2. Finding the value of x co-ordinate

Since the polar coordinates are$(6,\frac{2\pi }{3})$,, we can say that $r=6$and

$\begin{array}{l}\theta =\frac{2\pi }{3}\\ \Rightarrow x=r\cos \theta \\ \Rightarrow x=6\cos \left(\frac{2\pi }{3}\right)\\ \Rightarrow x=6\cdot \left(\frac{-1}{2}\right)\\ \Rightarrow x=-3\end{array}$

Step3. Finding the value of y co-ordinate

Now, to find the value of y,

$\begin{array}{l}\Rightarrow y=r\sin \theta \\ \Rightarrow y=6\sin \left(\frac{2\pi }{3}\right)\\ \Rightarrow y=6\cdot \left(\frac{\sqrt{3}}{2}\right)\\ \Rightarrow y=3\sqrt{3}\end{array}$