Question

Polar Coordinates to Rectangular Coordinates A point is graphed in polar form. Find its rectangular coordinates.

Short answer

The rectangular coordinates of the point $R(4,-\frac{2\pi }{3})$ are$-2,-2\sqrt{3}$ .

Step-by-step solution

Step 1.Given information.

From the figure, the polar coordinates of the point R are$4,-\frac{2\pi }{3}$ .

Step 2. We can find the rectangular coordinates of the point using the polar to rectangular coordinates conversion formulas.

From the given figure, we know that the polar coordinates of R are$4,-\frac{2\pi }{3}$Thus,$r=4$and$\theta =-\frac{2\pi }{3}\cdot$

We can use the conversion formulas$x=r\cos \theta$
and$y=r\sin \theta$.

To find the rectangular coordinates of the point.

$\begin{array}{l}\Rightarrow x=r\cos \theta \\ \Rightarrow x=4\cos (-\frac{2\pi }{3})\\ \Rightarrow x=4\cdot (-\frac{1}{2})\\ \Rightarrow x=-2\end{array}$

Now, to find the value of y,

$\begin{array}{l}\Rightarrow y=r\sin \theta \\ \Rightarrow y=4\sin (-\frac{2\pi }{3})\\ \Rightarrow y=(-\frac{\sqrt{3}}{2})\\ \Rightarrow y=-2\sqrt{3}\end{array}$