# Trigonometry/For Enthusiasts/Doing without Sine
## The Idea

We know that:
    $\sin ^{2}\theta =1-\cos ^{2}\theta \,$
So do we really need the $\sin \,$ function?
Or put another way, could we have worked out all our interesting formulas for things like $\cos(a+b)\,$ in terms just of $\cos \,$ and then derived every formula that has a $\sin \,$ in it from that?
The answer is yes.
We don't need to have one geometric argument for $\cos(a+b)\,$ and then do another geometric argument for $\sin(a+b)\,$. We could get our formulas for $\sin \,$ directly from formulas for $\cos \,$

## Angle Addition and Subtraction formulas

To find a formula for $\cos(\theta _{1}+\theta _{2})\,$ in terms of $\cos \left(\theta _{1}\right)$ and $\cos \left(\theta _{2}\right)$: construct two different right angle triangles each drawn with side $c\,$ having the same length of one, but with $\theta _{1}\neq \theta _{2}$, and therefore angle $\psi _{1}\neq \psi _{2}$. Scale up triangle two so that side $a_{2}\,$ is the same length as side $c_{1}\,$. Place the triangles so that side $c_{1}\,$ is coincidental with side $a_{2}\,$, and the angles $\theta _{1}\,$ and $\theta _{2}\,$ are juxtaposed to form angle $\theta _{3}=\theta _{1}+\theta _{2}\,$ at the origin. The circumference of the circle within which triangle two is embedded (circle 2) crosses side $a_{1}\,$ at point $g\,$, allowing a third right angle to be drawn from angle $\varphi _{2}$ to point $g\,$ . Now reset the scale of the entire figure so that side $c_{2}\,$ is considered to be of length 1. Side $a_{2}\,$ coincidental with side $c_{1}\,$ will then be of length $\cos \left(\theta _{1}\right)$, and so side $a_{1}\,$ will be of length $\cos \left(\theta _{1}\right)\cdot \cos \left(\theta _{2}\right)$ in which length lies point $g\,$. Draw a line parallel to line $a_{1}\,$ through the right angle of triangle two to produce a fourth right angle triangle, this one embedded in triangle two. Triangle 4 is a scaled copy of triangle 1, because:
The length of side $b_{4}\,$ is $cos(\varphi _{2})cos(\varphi _{4})=cos(\varphi _{2})cos(\varphi _{1})$ as $\theta _{4}\,=\theta _{1}\,$. Thus point $g\,$ is located at length:
$cos(\theta _{1}+\theta _{2})=cos(\theta _{1})cos(\theta _{2})-cos(\varphi _{1})cos(\varphi _{2}),$ where $\theta _{1}+\varphi _{1}=\theta _{2}+\varphi _{2}={\frac {\pi }{2}}$
giving us the "Cosine Angle Sum Formula".

### Proof that angle sum formula and double angle formula are consistent

We can apply this formula immediately to sum two equal angles:
From the theorem of Pythagoras we know that:
in this case:
Substituting into (I) gives:
which is identical to the "Cosine Double Angle Sum Formula":

## Pythagorean identity

Armed with this definition of the $sin\,$ function, we can restate the Theorem of Pythagoras for a right angled triangle with side c of length one, from:
to:
We can also restate the "Cosine Angle Sum Formula" from:
to:

## Sine Formulas

The price we have to pay for the notational convenience of this new function $sin\,$ is that we now have to answer questions like: Is there a "Sine Angle Sum Formula". Such questions can always be answered by taking the $cos\,$ form and selectively replacing $cos(\theta )^{2}\,$ by $1-sin(\theta )^{2}\,$ and then using algebra to simplify the resulting equation. Applying this technique to the "Cosine Angle Sum Formula" produces:
taking the square root of both sides produces the "Sine Angle Sum Formula"
We can use a similar technique to find the "Sine Half Angle Formula" from the "Cosine Half Angle Formula":
$cos\left({\frac {\theta }{2}}\right)={\sqrt {\frac {1+cos(\theta )}{2}}}$
We know that $1-cos\left({\frac {\theta }{2}}\right)^{2}=sin\left({\frac {\theta }{2}}\right)^{2}$, so squaring both sides of the "Cosine Half Angle Formula" and subtracting from one:
So far so good, but we still have a ${\frac {cos(\theta )}{2}}$ to get rid of. Use Pythagoras again to get the "Sine Half Angle Formula":
or perhaps a little more legibly as: