Question

In Exercises 32–47 convert the equations given in polar coordinates to rectangular coordinates.
$\theta =-\frac{\mathrm{\pi }}{6}$

Short answer

The required equation is$y=\frac{-x}{\sqrt{3}}$.

Step-by-step solution

Step 1. Given information.

The given equation in polar coordinates is:

$\theta =-\frac{\mathrm{\pi }}{6}$

Step 2. Find the equation in rectangular coordinates.

Use the formula $\tan \theta =\frac{y}{x}$,

$$\begin{gathered} \tan \left(-\frac{\mathrm{\pi }}{6}\right)=\frac{y}{x} \\ -\frac{1}{\sqrt{3}}=\frac{y}{x}\left[\mathrm{Since}\tan \left(-\frac{\mathrm{\pi }}{6}\right)=-\frac{1}{\sqrt{3}}\right] \\ -x=\sqrt{3}y \\ y=\frac{-x}{\sqrt{3}} \end{gathered}$$

Therefore, the equation in rectangular coordinates is$y=\frac{-x}{\sqrt{3}}$.