Equation of Straight Line in Plane/Normal Form/Proof 2

Proof

 * Straight-line-normal-form-Proof-2.png

Let $P = \tuple {x, y}$ be an arbitrary point on $\LL$.

Let $O$ be the origin of the Cartesian plane in which $\LL$ is embedded.

Let $PQ$ be the perpendicular dropped from $P$ to the $x$-axis.

Let $QS$ be the perpendicular dropped from $Q$ to the line $ON$.

Let $PR$ be the perpendicular dropped from $P$ to the line $QS$.

By definition of cosine:
 * $OS = OQ \cos \alpha$

By definition of sine:
 * $PR = PQ \sin \alpha$

Then: