Chapter 9, Problem 2CR

To determine

To determine: An equation for the line containing the origin that makes an angle of ${30}^{°}$ with the positive $x-$ axis.

Answer to Problem 2CR

Solution:

An equation for the line containing the origin that makes an angle of ${30}^{°}$ with the positive $x-$ axis is $y=\frac{\sqrt{3}}{3}x$ .

Explanation of Solution

Given information:

The line containing the origin that makes an angle of 30 ' with the positive $x$ - axis.

The line containing point $(0,0),$ making an angle of ${30}^{°}$ with the positive $x-$ axis has polar

equation $\theta =\frac{\pi }{6}$ .

Using the rectangular coordinates $\theta ={\tan }^{-1}\frac{y}{x}$

This implies that tan $\theta =\frac{y}{x}$ .

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

  $\Rightarrow \tan \left(\frac{\pi }{6}\right)=\frac{y}{x}$

since tan $\left(\frac{\pi }{6}\right)=\frac{\sqrt{3}}{3}$ ,

  $\Rightarrow \frac{\sqrt{3}}{3}=\frac{y}{x}$

  $\Rightarrow y=\frac{\sqrt{3}}{3}x$

Therefore, an equation for the line containing the ongin that makes anangle of ${30}^{°}$ with the positive $x-$ axis is $y=\frac{\sqrt{3}}{3}x$ .