Question

convert the equations given in polar coordinates to equations in rectangular coordinates

$\theta =0,\theta =\frac{\mathrm{\pi }}{4},\theta =-\frac{\mathrm{\pi }}{2}$

Short answer

The equations in rectangular form are:

$$\begin{gathered} y=0 \\ y=x \\ x=0 \end{gathered}$$

Step-by-step solution

Step 1. Given information

Equation in polar form

$\theta =0,\theta =\frac{\mathrm{\pi }}{4},\theta =-\frac{\mathrm{\pi }}{2}$

Step 2. Convert the equations in rectangular form.

As we know that in polar form

$\theta ={\tan }^{-1}\left(\frac{y}{x}\right)$

So,

localid="1652107972021" $$\begin{gathered} \theta =0 \\ {\tan }^{-1}\left(\frac{y}{x}\right)=0 \\ \frac{y}{x}=\tan 0 \\ \frac{y}{x}=0 \\ y=0 \end{gathered}$$

$$\begin{gathered} \theta =\frac{\mathrm{\pi }}{4} \\ {\tan }^{-1}\left(\frac{y}{x}\right)=\frac{\mathrm{\pi }}{4} \\ \frac{y}{x}=\tan \frac{\mathrm{\pi }}{4} \\ \frac{y}{x}=1 \\ y=x \end{gathered}$$

$$\begin{gathered} \theta =-\frac{\mathrm{\pi }}{2} \\ {\tan }^{-1}\left(\frac{y}{x}\right)=-\frac{\mathrm{\pi }}{2} \\ \frac{y}{x}=\tan \left(-\frac{\mathrm{\pi }}{2}\right) \\ \frac{y}{x}=-\frac{1}{0} \\ x=0 \end{gathered}$$