Cosine of Angle in Cartesian Plane

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Theorem

Let $P = \left({x, y}\right)$ be a point in the cartesian coordinate plane whose origin is at $O$.

Let $\theta$ be the angle between the $x$-axis and the line $OP$.

Let $r$ be the length of $OP$.

Then:

$\cos \theta = \dfrac x r$

where $\cos$ denotes the cosine of $\theta$.


Proof

CosineCartesian.png

Let a unit circle $C$ be drawn with its center at the origin $O$.

Let $Q$ be the point on $C$ which intersects $OP$.

From Parallelism implies Equal Alternate Interior Angles, $\angle OQR = \theta$.

Thus:

$(1): \quad \cos \theta = RQ$

by definition of cosine


$\angle OSP = \angle ORQ$, as both are right angles.

Both $\triangle OSP$ and $\triangle ORQ$ share angle $\angle SOP$.

By Triangles with Two Equal Angles are Similar it follows that $\triangle OSP$ and $\triangle ORQ$ are similar.

By definition of similarity:

Then:

\(\displaystyle \frac x r\) \(=\) \(\displaystyle \frac {SP} {OP}\)
\(\displaystyle \) \(=\) \(\displaystyle \frac {RQ} {OQ}\) by definition of similarity
\(\displaystyle \) \(=\) \(\displaystyle RQ\) as $OP$ is the radius of the unit circle
\(\displaystyle \) \(=\) \(\displaystyle \cos \theta\) from $(1)$ above

$\blacksquare$


CosineCartesianSecond.png

When $\theta$ is obtuse, the same argument holds, except that this time both $x$ and $\cos \theta$ are negative.


When $\theta = \dfrac \pi 2$ we have that $x = 0$ and $\cos \theta = 0 = \dfrac x r$.

Thus $y = r$ and $\sin \theta = 1 \dfrac y r$.

Thus the relation holds for $\theta = \dfrac \pi 2$.


When $\pi < \theta < 2 \pi$ the diagram can be reflected in the $x$-axis.

Thus the relation continues to hold.


When $\theta = 0$ we have that $y = 0$ and $\cos \theta = 1 = \dfrac x r$.

When $\theta = \pi$ we have that $y = 0$ and $x = -r$, while $\cos \theta = -1 = \dfrac x r$.

Hence the result.

$\blacksquare$


Sources