Why can $(x,y)$ coordinates be defined as $(\cos\theta,\sin\theta)$ for the unit circle?

Question

I am having a hard time understanding that how/why can we define $(x,y)$ as $(\cos\theta,\sin\theta$. I googled and found out that we can do that because for $\theta\geq\frac\pi2$, a triangle can be contructed in all cases, so we can always get a $\frac{\text{opposite}}{\text{hypotenuse}}$ relationship. But what I cannot understand is that:
Suppose $\theta$ is in the second quadrant. So the triangle we will contruct will have a $\frac{\text{opposite}}{\text{hypotenuse}}$ relationship for $180-\theta$ not $\theta$. So how can we define it like that?
I know it seems a bit crazy but lets put it like this. $\sin\theta$ is opposite upon hypotenuse. But suppose the angle is obtuse, then how will we get this relationship?

Answer 1

When one first meets the trigonometric functions, they are best interpreted as ratios of different sides of a right triangle. Later, one might ask the question: what is the sine of 91°, for example? There is a German article on Wikipedia: https://de.wikipedia.org/wiki/Permanenzprinzip. I don't know whether a similar term exists in English, but it is a principle followed in generalizations. It is also applied in the unit circle-definition of the trigonometric functions. When the angle is between 0 and 90°, we get back the previous definitions. It also leads to the polar coordinate system, where the minus sign is never negligible, and the sine-cosine interpretation should give back the x-y components.

Another way to approach: how would you extend the definitions of sine and cosine to any real angle?

Nonetheless, it might worth trying different definitions and see their implications.

The further generalization is the function itself: one can easily talk about the sine, cosine, etc. functions without any reference to the unit circle: their representation as Taylor-series, and finally (?) the Euler-form, relating the exponential to the trigonometric function. This generalization seems to be a very effective one.

(If you don't know German or Hungarian: the linked "Permanenzprinzip" means the principle that a generalization should be one that has as many similarities with the original definitions as possible. The first instance one encounters is the idea of number systems and algebraic operations.)

Answer 2

The usual (most general and straightforward) geometric definition of the sine and cosine is:

$(\cos \theta,\sin \theta)$ are the coordinates of the point on the unit circle whose arc distance from $(1,0)$ measured counterclockwise along the circle itself, is $\theta$.

This immediately gives the right value for all $\theta$ -- where "right" means, among a lot of other things, that it is the only choice that makes $\sin$ and $\cos$ analytic everywhere and agrees with the right-triangle definition you seem to prefer for angles between $0$ and $\pi/2$.

In analysis different definitions are often preferred, usually the power series:

$$ \cos x = 1 - \frac{x^2}{2} + \frac{x^4}{24} - \cdots + (-1)^n \frac{x^{2n}}{(2n)!} + \cdots \\ \sin x = x - \frac{x^3}{6} + \frac{x^5}{120} - \cdots + (-1)^n \frac{x^{2n+1}}{(2n+1)!} + \cdots $$

which have the advantage that they're easy to generalize to complex arguments. But it then takes a fair amount of proofwork to show that these series converge for every $x$, and that these definitions agree with the geometric ones.