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 coordinate plane.


SineFirstQuadrant.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 $x$-axis.


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


Definition from Real Numbers

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

\(\displaystyle \sin x\) \(=\) \(\displaystyle \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!}\)
\(\displaystyle \) \(=\) \(\displaystyle x - \frac {x^3} {3!} + \frac {x^5} {5!} - \cdots\)


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:

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

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:

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

That is:

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

$\blacksquare$

Definition from Circle equivalent to Definition from Real Numbers

Let, $\sin_G$ be the Geometric Sine from Definition:Sine/Definition from Circle. $\arcsin_G$ is the inverse of this function.

Let $sin_A$ be the analytic sine function for real numbers, the one defined by Definition:Sine/Real Numbers. $\arcsin_A$ is the inverse of this function.

We know from Arcsin as an Integral that $\arcsin_A$ and $\arcsin_G$ are the same function.

$x=\sin_A\left({\theta}\right) \iff \arcsin_A\left(x\right)=\theta \iff \arcsin_G\left(x\right)=\theta \iff x=\sin_G\left({\theta}\right)$

$\blacksquare$