Chapter 8.1, Problem 52E

To determine

To convert the given polar equation to rectangular coordinates

  $\theta =\pi$

Answer to Problem 52E

The rectangular equation is $y=0$ .

Explanation of Solution

Given: Polar equation: $\theta =\pi$

Formula Used:

A polar equation is any equation that describes a relation between r and θ , where r represents the distance from the pole (origin) to a point on a curve, and θ represents the counter-clockwise angle made by a point on a curve, the pole, and the positive x-axis.

Also, $x=r\cos \theta$ and $y=r\sin \theta$

  $r=\sqrt{x^2+y^2}$ and $\theta ={\tan }^{-1}\left(\frac{y}{x}\right)$

  $\tan \left(-\theta \right)=-\tan \theta$

  $\tan \pi =0$

Calculation:

Given Polar equation is $\theta =\pi$

Using the above definition, we have:

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

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

  $\begin{array}{l}\tan \left(\pi \right)=\frac{y}{x}\\ 0=\frac{y}{x}\\ y=0\end{array}$

Thus, the rectangular equation is $y=0$ .

Conclusion:

The rectangular equation is $y=0$ .