Chapter 9.2, Problem 81AYU

Show that the graph of the equation $r\sin \theta \text{ = }a$ is a horizontal line $a$ units above the pole if $a\ge \text{ 0 and }\left|a\right|$ units below the pole if $a<\text{ 0}$.

To determine

To show: The graph of the polar equation $r\text{sin}\theta = a$ is a horizontal line $a$ units above the pole if $a\ge 0$ and $\left|a\right|$ units below the pole if $a<0$.

Explanation of Solution

Given:

It is asked to show that the graph of the polar equation $r\text{sin}\theta = a$ is a horizontal line $a$ units above the pole if $a\ge 0$ and $\left|a\right|$ units below the pole if $a<0$.

It is a known fact that each point in the polar coordinate system is given by $\left(r,\theta \right)$ where $r$ is the distance from the pole (origin) to the point, and $\theta$ is the counter clockwise angle that is made with the point pole, and the positive $x$-axis.

To convert back and forth between polar and rectangular coordinates, we have the following formulas:

$\begin{array}{l}\text{cos}\theta = \frac{x}{r}\hfill \\ x = r\text{cos}\theta \hfill \\ \text{sin}\theta = \frac{y}{r}\hfill \\ y = r\text{sin}\theta \hfill \\ r^2 = {\left(r\text{cos}\theta \right)}^2 + {\left(r\text{sin}\theta \right)}^2 = x^2 + y^2\hfill \\ \frac{\text{sin}\theta }{\text{cos}\theta } = \frac{\frac{y}{r}}{\frac{x}{r}}\hfill \\ \tan \theta = \frac{y}{x}\hfill \end{array}$

And $\left(x,y\right) = \left(r\text{cos}\theta ,r\text{sin}\theta \right)$

Consider $r\text{sin}\theta = a$,

Converting to rectangular coordinate, take $r\text{sin}\theta = y$. The equation becomes $y = a$.

In rectangular coordinates, $y = a$ is a horizontal equation $a$ units above the $x$-axis if $a\ge 0$ and $\left|a\right|$ units below the $x$-axis if $a<0$.

In polar coordinates, $r\text{sin}\theta = a$ is a horizontal equation $a$ units above the pole if $a\ge 0$ and $\left|a\right|$ units below the pole if $a<0$.

Hence proved.