Equivalence of Definitions of Sine of Angle

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Theorem

Let $\theta$ be an angle.

The following definitions of the concept of sine of $\theta$ are equivalent:

Definition from Triangle

SineCosine.png

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

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

Definition from Circle

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


SineFirstQuadrant.png


Let $P = \tuple {x, y}$ 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 $x$-axis.


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

Hence in the first quadrant, the sine is positive.


Definition from Real Numbers

The real function $\sin: \R \to \R$ is defined as:

\(\ds \forall x \in \R: \, \) \(\ds \sin x\) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!}\)
\(\ds \) \(=\) \(\ds x - \frac {x^3} {3!} + \frac {x^5} {5!} - \cdots\)


Proof

Definition from Triangle implies Definition from Circle

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

Consider the triangle $\triangle OAP$.

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


Thus:

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

That is:

$\sin \theta = AP$

$\Box$


Definition from Circle implies Definition from Triangle

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

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

We have that:

$\angle CAB = \angle POA = \theta$
$\angle ABC = \angle OAP$ 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:

\(\ds \frac {\text{Opposite} } {\text{Hypotenuse} }\) \(=\) \(\ds \frac {BC} {AC}\) by definition
\(\ds \) \(=\) \(\ds \frac {AP} {OP}\) Definition of Similar Triangles
\(\ds \) \(=\) \(\ds AP\) as $OP$ is the radius of the unit circle
\(\ds \) \(=\) \(\ds \sin \theta\) by definition

That is:

$\dfrac {\text{Opposite} } {\text{Hypotenuse} } = \sin \theta$

$\Box$


Definition from Circle equivalent to Definition from Real Numbers

This is demonstrated in Arcsine as Integral.

$\blacksquare$