Equivalence of Definitions of Cosine of Angle

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Theorem

Let $\theta$ be an angle.


The following definitions of the concept of cosine are equivalent:

Definition from Triangle

SineCosine.png

In the above right triangle, we are concerned about the angle $\theta$.

The cosine of $\angle \theta$ is defined as being $\dfrac {\text{Adjacent}} {\text{Hypotenuse}}$.

Definition from Circle

Consider a unit circle $C$ whose center is at the origin of a cartesian coordinate plane.


CosineFirstQuadrant.png


Let $P = \left({x, y}\right)$ be the point on $C$ in the first quadrant such that $\theta$ is the angle made by $OP$ with the $x$-axis.

Let $AP$ be the perpendicular from $P$ to the $y$-axis.


Then the cosine of $\theta$ is defined as the length of $AP$.



Proof

Definition from Triangle implies Definition from Circle

Let $\cos \theta$ be defined as $\dfrac {\text{Adjacent}} {\text{Hypotenuse}}$ in a right triangle.

Consider the triangle $\triangle OAP$.

By construction, $\angle OAP$ is a right angle.

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


Thus:

\(\displaystyle \cos \theta\) \(=\) \(\displaystyle \frac {AP} {OP}\)
\(\displaystyle \) \(=\) \(\displaystyle \frac {AP} 1\) as $OP$ is the radius of the unit circle
\(\displaystyle \) \(=\) \(\displaystyle AP\)

That is:

$\cos \theta = AP$

$\Box$


Definition from Circle implies Definition from Triangle

Let $\cos \theta$ be defined as the length of $AP$ in the triangle $\triangle OAP$.

Compare $\triangle OAP$ with $\triangle ABC$ in the diagram above.

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

We have that:

$\angle CAB = \angle OPA = \theta$
$\angle ABC = \angle OAP$ which is a right angle

Therefore by Triangles with Two Equal Angles are Similar it follows that $\triangle OAP$ and $\triangle ABC$ are similar.

By definition of similarity:

\(\displaystyle \frac {\text{Adjacent} } {\text{Hypotenuse} }\) \(=\) \(\displaystyle \frac {AB} {AC}\) by definition
\(\displaystyle \) \(=\) \(\displaystyle \frac {AP} {OP}\) by definition of similarity
\(\displaystyle \) \(=\) \(\displaystyle AP\) as $OP$ is the radius of the unit circle
\(\displaystyle \) \(=\) \(\displaystyle \cos \theta\) by definition

That is:

$\dfrac {\text{Adjacent} } {\text{Hypotenuse} } = \cos \theta$

$\blacksquare$