Fundamental Theorem of Trigonometry

Question

This is a pretty open ended question and I apologize, in advance, if this is not the place for it. But what do you recommend should be given the title of the Fundamental Theorem of Trigonometry and why? Should we have to restrict ourselves to the planar case... I think so.

Answer 1

$$\boxed{\sin^2x+\cos^2x=1.}$$

Answer 2

$e^{i\theta}=\cos \theta + i \sin \theta$.

You can derive a bunch of the relations from here!

Answer 3

Fundamental theorem, imho, would be:

A magnifying glass that increases the size of an object $k$ times:
1) Doesn't change angles
2) Increases length by a factor of $k$

From this you can (informally) derive the existence of sine, cosine, $\pi$, the $k^2$ increase in area, figure out problems of similar triangles, etc.

Answer 4

$$\frac{\sin x}{\cos x} = \tan x$$ $$sin^2 x + cos^2x = 1$$

Answer 5

The identity $\sin^2x+\cos^2x=1$ comes from Pythagoras.

I think the fact that $\sin x$ and $\cos x$ (for a right-angled triangle) are well defined at all is the fundamental theorem.

Fundamental Theorem of Trigonometry The ratio between corresponding sides of similar triangles are equal.

Edit: Daniel V has a similar idea.