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### Precalculus Mathematics for Calculus

Book edition 7th Edition
Author(s) James Stewart, Lothar Redlin, Saleem Watson
Pages 948 pages
ISBN 9781337067508

# Rectangular Coordinates to Polar Coordinates. A point is graphed in rectangular form. Find polar coordinates for the point, with $r>0$and $0<\theta <2\pi$

The polar coordinates are$3\sqrt{2},\frac{3\pi }{4}$

See the step by step solution

## Step 1. Given information

A point in the rectangular coordinate system

## Step 2. Concept used

Four points match the polar coordinates of a point. The radius of each point is four units. Their angles are the only thing that separates them. As a result, find the point that has the same angle as the specified point..

## Step 3. Calculation

The coordinates of the point are. So,$x=-3$ and $y=3$. Now convert it into polar coordinates.

Use the formulae as-

$\begin{array}{l}{r}^{2}={x}^{2}+{y}^{2}\\ \mathrm{tan}\theta =\frac{y}{x}\end{array}$

$\begin{array}{l}⇒{r}^{2}={x}^{2}+{y}^{2}\\ ⇒{r}^{2}\text{Invalid element}\text{Invalid element}=+\\ ⇒{r}^{2}=18\\ ⇒r=±\sqrt{18}or±3\sqrt{2}\end{array}$

But they had given,$r>0$

Therefore, $r=+3\sqrt{2}$

The value of angle can be determined as-

$\begin{array}{l}⇒\mathrm{tan}\theta =\frac{y}{x}\\ ⇒\mathrm{tan}\theta =\frac{3}{-3}\\ ⇒\mathrm{tan}\theta =-1\\ ⇒\theta ={\mathrm{tan}}^{-1}\left(-1\right)\\ ⇒\theta =\frac{3\pi }{4},\frac{7\pi }{4}\end{array}$

But in the question,$0<\theta <2\pi$ . Therefore $\theta =\frac{3\pi }{4}$

Polar coordinates of the point are $3\sqrt{2},\frac{3\pi }{4}$
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