Chapter 9.5, Problem 59E

To determine

To convert the polar equation to rectangular form.

Answer to Problem 59E

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

Explanation of Solution

Given information: The polar equation $\theta =\frac{5\pi }{6}$ .

Calculation: The required rectangular form from polar equation is obtained as,

  $\theta =\frac{5\pi }{6}$

Taking tan both sides,

  $\begin{array}{l}\tan \theta =\tan \frac{5\pi }{6}\\ \frac{\sin \theta }{\cos \theta }=\frac{-1}{\sqrt{3}}\end{array}$

Multiply both side by r.

  $\begin{array}{l}r.\frac{\sin \theta }{\cos \theta }=\frac{-r}{\sqrt{3}}\\ r\sin \theta =\frac{-r}{\sqrt{3}}\cos \theta \end{array}$

Use the following substitution:

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

  $\begin{array}{l}r\sin \theta =\frac{-r}{\sqrt{3}}\cos \theta \\ y=\frac{-x}{\sqrt{3}}\end{array}$

Hence, the required rectangular form is $y=\frac{-x}{\sqrt{3}}$ .