Q.Find the equation of the line, the length of whose perpendicular from the origin is $5$ units, and the perpendicular makes an angle of $30^\circ$ with the positive x-axis.

The perpendicular length from the origin and its direction call for the normal form $x\cos\alpha + y\sin\alpha = p$.

The normal (perpendicular) form of a line, where $p$ is the length of the perpendicular from the origin and $\alpha$ the angle that perpendicular makes with the positive x-axis, is

Here $p = 5$ and $\alpha = 30^\circ$, with $\cos 30^\circ = \dfrac{\sqrt{3}}{2}$ and $\sin 30^\circ = \dfrac{1}{2}$:

Multiply through by $2$:

$\sqrt{3}\,x + y = 10 \ \Rightarrow\ \sqrt{3}\,x + y - 10 = 0$ …

Swapping $\cos$ and $\sin$, or using $\sin 30^\circ = \tfrac{\sqrt3}{2}$ (it is $\tfrac12$). Recall the exact values $\cos 30^\circ = \tfrac …