Question

In Exercises 32–47 convert the equations given in polar coordinates to rectangular coordinates.
$\theta =\frac{7\mathrm{\pi }}{6}$

Short answer

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

Step-by-step solution

Step 1. Given information.

The given equation in polar coordinates is:

$\theta =\frac{7\mathrm{\pi }}{6}$

Step 2. Find the equation in rectangular coordinates.

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

$$\begin{gathered} \tan \left(\frac{7\mathrm{\pi }}{6}\right)=\frac{y}{x} \\ \frac{\sqrt{3}}{3}=\frac{y}{x}\left[\tan \frac{7\mathrm{\pi }}{6}=\frac{\sqrt{3}}{3}\right] \\ 3y=\sqrt{3}x \\ y=\frac{\sqrt{3}x}{3} \end{gathered}$$

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