Characterisation of Sine and Cosine

Theorem
The definitions for sine and cosine are equivalent.

That is:


 * $\displaystyle \sin x = \sum_{n \mathop = 0}^\infty \left({-1}\right)^n \frac {x^{2n+1}}{\left({2n+1}\right)!} \iff \sin x = \frac {\text{Opposite}} {\text{Hypotenuse}}$


 * $\displaystyle \cos x = \sum_{n \mathop = 0}^\infty \left({-1}\right)^n \frac {x^{2n}}{\left({2n}\right)!} \iff \cos x = \frac {\text{Adjacent}} {\text{Hypotenuse}}$

Proof
Let $s \left({x}\right): \R \to \R$, $c \left({x}\right): \R \to \R$ be two functions that satisfy:
 * 1) $s' \left({x}\right)=c \left({x}\right)$
 * 2) $c' \left({x}\right)=-s \left({x}\right)$
 * 3) $s(0)=0$
 * 4) $c(0)=1$
 * 5) $\forall x : s^2 \left({x}\right)+c^2 \left({x}\right)=1$

Let $f \left({x}\right): \R \to \R$, $g \left({x}\right): \R \to \R$ also be two functions that satisfy:
 * 1) $f' \left({x}\right)=g \left({x}\right)$
 * 2) $g' \left({x}\right)=-f \left({x}\right)$
 * 3) $f(0)=0$
 * 4) $g(0)=1$
 * 5) $\forall x : f^2 \left({x}\right)+g^2 \left({x}\right)=1$

We will show that $f \left({x}\right)=s \left({x}\right)$ and $c \left({x}\right)=g \left({x}\right)$.

Define: $h \left({x}\right)=(c \left({x}\right)-g \left({x}\right))^2+(s \left({x}\right)-f \left({x}\right))^2$.

Notice that $h \left({x}\right)=0$ for all $x$ iff $c \left({x}\right)=g \left({x}\right)$ and $s \left({x}\right)=f \left({x}\right)$ for all $x$.

Then:

By taking $h' \left({x}\right)$:

So that $h \left({x}\right)$ is a constant function ($h \left({x}\right)=k$).

Also:

Since $h \left({x}\right)$ is constant, then $h \left({x}\right)=0$ for all $x$. Then $c \left({x}\right)=g \left({x}\right)$ and $s \left({x}\right)=f \left({x}\right)$.

Since, by Derivative of Sine Function,  Derivative of Cosine Function,  Sine of Zero is Zero,  Cosine of Zero is One and  Sum of Squares of Sine and Cosine both definitions satisfy all these properties, then they must be the same.