Question

In Problems 39-54,the poplar coordinates of a point are given.Find rectangular coordinates of each point.

$(-1,-\frac{\mathrm{\pi }}{3})$

Short answer

The rectangular coordinates corresponding to the polar coordinates $(-1,-\frac{\mathrm{\pi }}{3})$are $(-\frac{1}{2},\frac{\sqrt{3}}{2})$.

Step-by-step solution

Step 1. Given information

The given poplar coordinates of a point$(-1,-\frac{\mathrm{\pi }}{3})$

Step 2. Finding the value of x.

For a point P on the terminal side of $\theta$and having polar coordinates $(r,\theta )$, the corresponding rectangular coordinates are given by $x=r\cos \theta$and$y=r\sin \theta$.
Substitute $-1$ for 1 and $-\frac{\mathrm{\pi }}{3}$for $\theta$in $x=r\cos \theta$ to get the value of x.

$$\begin{gathered} x=r\cos \theta \\ =-1.\cos (-\frac{\mathrm{\pi }}{3}) \\ =-1.\frac{1}{2} \\ =-\frac{1}{2} \end{gathered}$$

Step 3. Finding the value of y.

Substitute $-1$ for r and $-\frac{\mathrm{\pi }}{3}$for $\theta$in to get the value of y.

$$\begin{gathered} y=r\sin \theta \\ =-1.\sin (-\frac{\mathrm{\pi }}{3}) \\ =-1.-\sin \left(\frac{\mathrm{\pi }}{3}\right) \\ =-1.(-\frac{\sqrt{3}}{2}) \\ =\frac{\sqrt{3}}{2} \end{gathered}$$

Therefore, the rectangular coordinates corresponding to the polar coordinates$(-1,-\frac{\mathrm{\pi }}{3})$are$(-\frac{1}{2},\frac{\sqrt{3}}{2})$.