Question

For questions 1-5, the polar coordinates of a point are given. Find the rectangular coordinates of each point.

Answer 1

QUESTION 1

We want to convert $(5,\frac{\pi}{4})$ from polar coordinates to rectangular coordinates.

We use the formula;

$x=r\cos \theta$

$y=r\sin \theta$

We plug in $r=5$ and $\theta=\frac{\pi}{4}$ to obtain;

$x=5\cos \frac{\pi}{4}=\frac{5\sqrt{2}}{2}$

$y=5\sin \frac{\pi}{4}=\frac{5\sqrt{2}}{2}$

The rectangular coordinate is $(\frac{5\sqrt{2}}{2},\frac{5\sqrt{2}}{2})$

QUESTION 2

We want to convert $(-2,\frac{\pi}{6})$ from polar coordinates to rectangular coordinates.

We use the formula;

$x=r\cos \theta$

$y=r\sin \theta$

We plug in $r=-2$ and $\theta=\frac{\pi}{6}$ to obtain;

$x=-2\cos \frac{\pi}{6}=-1$

$y=-2\sin \frac{\pi}{6}=-\sqrt{3}$

The rectangular coordinate is $(-1,-\sqrt{3})$

QUESTION 3

We want to convert $(-1,\frac{-2\pi}{3})$ from polar coordinates to rectangular coordinates.

We use the formula;

$x=r\cos \theta$

$y=r\sin \theta$

We plug in $r=-1$ and $\theta=\frac{-2\pi}{3}$ to obtain;

$x=-1\cos \frac{-2\pi}{3}=\frac{1}{2}$

$y=-1\sin \frac{-2\pi}{3}=\frac{\sqrt{3}}{2}$

The rectangular coordinate is $(\frac{1}{2},\frac{\sqrt{3}}{2})$

QUESTION 4

We want to convert $(3,225\degree)$ from polar coordinates to rectangular coordinates.

We use the formula;

$x=r\cos \theta$

$y=r\sin \theta$

We plug in $r=3$ and $\theta=225\degree$ to obtain;

$x=3\cos (225\degree)=-3\frac{\sqrt{2}}{2}$

$y=3\sin(225\degree)=-\frac{3\sqrt{2}}{2}$

The rectangular coordinate is $(-\frac{3\sqrt{2}}{2},-3\frac{\sqrt{2}}{2})$

QUESTION 5

We want to convert $(-3,330\degree)$ from polar coordinates to rectangular coordinates.

We use the formula;

$x=r\cos \theta$

$y=r\sin \theta$

We plug in $r=-3$ and $\theta=330\degree$ to obtain;

$x=-3\cos (330\degree)=-\frac{3\sqrt{3}}{2}$

$y=-3\sin(330\degree)=\frac{3}{2}$

The rectangular coordinate is $(-\frac{3\sqrt{3}}{2},\frac{3}{2})$