Question

Find a polar equation for the line containing the origin that makes an angle of 30 degree with the positive x-axis.

Short answer

The equation is$\theta =n\mathrm{\pi }+\frac{\mathrm{\pi }}{6}$

Step-by-step solution

Step 1. Given Information

The given data isthe line containing the origin that makes an angle of 30 degree with the positive x-axis.

Step 2. Explanation

The equation of a line in slope-form in rectangular Cartesian coordinates is given below, $y=mx+c$

The line passes through the origin of the coordinate plane and has a slope equal to $m=\tan 30°$

The value of c must be obtained first to determine the equation of line.

$$\begin{gathered} 0=\tan (30°)\left(0\right)+c \\ c=0 \end{gathered}$$

Thus, the equation of line in rectangular cartesian coordinates is $x=\sqrt{3}y$

The equation can be obtained in polar form by substituting $x=r\cos \theta ,y=r\sin \theta$, we get,

$$\begin{gathered} r\cos \theta =\sqrt{3}\left(r\sin \theta \right) \\ \tan \theta =\frac{1}{\sqrt{3}} \end{gathered}$$

Step 3. Calculation

Now find the value of angle in radian.

$$\begin{gathered} \tan \theta =\frac{1}{\sqrt{3}} \\ \tan \theta =\tan \frac{\mathrm{\pi }}{6} \\ \mathrm{As}\mathrm{we}\mathrm{know}\mathrm{that}\mathrm{if}\tan \theta =\tan \alpha \mathrm{then}\theta =n\mathrm{\pi }+\mathrm{\alpha } \\ \Rightarrow \mathrm{\theta }=\mathrm{n\pi }+\frac{\mathrm{\pi }}{6} \end{gathered}$$