Question

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

$\left(-2,\frac{2\mathrm{\pi }}{3}\right)$

Short answer

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

Step-by-step solution

Step 1. Given information

The given poplar coordinates of a point$(-2,\frac{2\mathrm{\pi }}{3})$

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 $-2$ for r and $\frac{2\mathrm{\pi }}{3}$for $\theta$in $x=r\cos \theta$ to get the value of x.

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

step 3. Finding the value of y.

Substitute $-2$ for r and $\frac{2\mathrm{\pi }}{3}$for $\theta$in $y=r\sin \theta$to get the value of $y$.

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

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