Chapter 8.2, Problem 63E

To determine

To calculate: To compare the rectangular equation of the line $y=2$ with its polar equation and find which coordinate system is simpler and to find which coordinate system one would choose to study lines.

Answer to Problem 63E

Rectangular coordinate system is simpler and easier to study lines

Explanation of Solution

Given: Rectangular equation of line is $y=2$

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.

Calculation:

Given rectangular equation of line as follows:

  $y=2$

Polar equation is given as

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

Now, $\theta$ is calculated as follows:

  $\tan \theta =\frac{y}{x}$

Substituting the values to get the value of $\theta$ :

  $\begin{array}{l}\tan \theta =\frac{2}{x}\\ \theta ={\tan }^{-1}\frac{2}{x}\end{array}$

Now, $x$ can take any value on the real line.

In particular, $x$ can take value $0$ which makes the entire inverse tangent function undefined.

Hence, rectangular coordinate system is much simpler to solve as well as study the lines.

Conclusion:

Hence, Rectangular coordinate system is simpler and easier to study lines