Prove the geometric significance of $sinx$ and $cosx$

Question

bobo-i-bo · Answer

I have proved it one way, but was wondering what you guys could come up with:
Let the $sinx$ and $cosx$ be defined by their standard Maclaurin series and you may assume any analytic properties of the trignometric functions. Can you prove that $sinx$ and $cosx$ then form the sides of a right angled triangle and (the most amazing part) that x is one of the angles of the triangle?

reemii · Answer

Suppose $a^2+b^2=1$, can you conclude that $a,b$, and 1 are the measure of the three sides of a right angle? If the answer is yes, then, yes. Because you said we may assume that $sin^2x+cos^2x = 1$ holds.

reemii · Answer

right triangle*

bobo-i-bo · Answer

Yup, you can assume Pythagoras Theorem, and also by the analytic definition of cos and sin, we can prove $sin^2x+cos^2x=1$. So how would you prove that the angle is $x$ @reemii ?