Question

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

Short answer

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

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-

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

$\begin{array}{l}\Rightarrow r^2=x^2+y^2\\ \Rightarrow r^2\text{Invalid <m:msup> element}\text{Invalid <m:msup> element}=+\\ \Rightarrow r^2=18\\ \Rightarrow r=\pm \sqrt{18}or\pm 3\sqrt{2}\end{array}$

But they had given,$r>0$

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

The value of angle can be determined as-

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

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

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