Why is the cos, sin definition of the unit circle true?

Question

So imagine we have a unit circle and there a point $M$ on it. Then the $x$ coordinate of $M$ is $\cos(\theta) $ ($\theta$ is the angle ${IOM}$ as you know) and its $y$ coordinate is $\sin(\theta)$.

But why is this true? Why isn't it the inverse?

Answer 1

I suppose you got this definition:

The number $\cos \theta$ is defined as the ratio $IO/OM$ of a triangle $IOM$ which has a right angle at $I$ and angle $\theta$ at $O$.

Now if you take a unit circle in an orthogonal coordinate system centered in $O$ and take a point $M$ on the cirlce and project the point $M$ orthogonally on the line of the $x$-axis, you get a point $I$ such that the triangle $IOM$ has the properties stated above. In particular being $OM=1$ you get that $OI= \cos \theta$ and $OI$ is actually defined as the $x$ coordinate of the point $M$.