Background
In school they teach that: $$sin(\theta) = \frac{opposite}{hypotenuse}$$
I'm confused because the argument $\theta$ is not in the RHS. It almost feels like the expression should be: $$sin(\theta, opposite,hypotenuse) = \theta*0+\frac{opposite}{hypotenuse}$$
Question 1:
Can you clarify why the LHS and RHS of $sin(\theta) = o/h$ reference totally different variables?
I'm looking for a "nice" formula for the function $sin(\theta)$. For example,the formula for distance is "nice": $d(x,y) = \sqrt{x^2 + y^2}$ and this is a formula that makes total sense to me.
I know about this approximation if $x$ is in radians: $sin(x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!} x^{2n+1}$
Question 2:
Is it correct to say there is no "nice" function of sine? Why not?
[Edit: My own definition of "nice" is very arbitrary and not precise. If you were a beginner again of trigonometry, how would you explain what $sin(\theta)$ is in terms of $\theta$ only?]
Thank you for your help and patience!!
Question 1: if you want $\theta$ to appear on the right-hand side, I think the following is much more natural, but it's completely unhelpful.
Write $\mathrm{opp}(\theta, t)$ for the length of the side opposite from the angle $\theta$ when the length of the adjacent side in a right-angled triangle is $t$.
Write $\mathrm{hyp}(\theta, t)$ for the length of the hypotenuse when the length of the adjacent side is $t$.
Then $$\sin(\theta) = \frac{\mathrm{opp}(\theta, t)}{\mathrm{hyp}(\theta, t)}$$ for any $t > 0$.