Question

Explain how you proceed to convert from rectangular coordinates to polar coordinates.

Short answer

To convert from rectangular coordinates to polar coordinates use the formula$r=\sqrt{x^2+y^2}and\theta ={\tan }^{-1}\frac{y}{x}.$

Step-by-step solution

Step 1. Given Information

The points $\left(r,\theta \right)$are determined by their angle on the unit circle and their distance from the origin in the polar coordinates.

The points $\left(x,y\right)$ are determined by their distance from the xand yaxes in the rectangular coordinates.

Step 2. Explaining

As we know if a point Pwith polar coordinates are $\left(r,\theta \right)$, the rectangular coordinates $\left(x,y\right)$ of Pare given by $$\begin{gathered} x=r\cos \theta \\ y=r\sin \theta \end{gathered}$$

Now, square and add both equations

$$\begin{gathered} x^2+y^2={\left(r\cos \theta \right)}^2+{\left(r\sin \theta \right)}^2 \\ {x}^2+y^2=r^2\left({\cos }^2\theta +{\sin }^2\theta \right) \\ {x}^2+y^2=r^2\left[{\cos }^2\theta +{\sin }^2\theta =1\right] \\ r=\sqrt{x^2+y^2} \end{gathered}$$

And as we know

$$\begin{gathered} \tan \theta =\frac{y}{x} \\ So,\theta ={\tan }^{-1}\frac{y}{x} \end{gathered}$$