Why is the sine and cosine always between $-1$ and $1$? Why is the sine and cosine always between $-1$ and $1$? If I would have circle with a radius other than $1$, then it wouldn't be between $-1$ and $1$ anymore, would it?
This also ties in with another thing I'm getting confused about: the $x$ and $y$-coordinates of a point on a circle are defined by $\cos\theta$ and $\sin\theta$. But what if the radius would be defined as anything other than having a length of $1$ unit? Would you still be able to find the $(x, y)$ coordinate of the point on the circle?
 A: Short answer: The sine function takes values from $-1$ to $1$ because that's just the nature of the function.

Longer answer:
The elementary definition of the sine function for angles from $0$ to $\pi/2$ is that it is the ratio of the length of one (opposite to the corner of which the sine is calculated) catete of a right angle triangle to the hypothenuse of the triangle, and since the hypothenuse is always the longest side of a right angle triangle, the sine of an angle will always be smaller than $1$ (and, for this definition, larger than $0$).
The more general definition of the function is that the sine of an angle is equal to the height of the point on the unit circle given the correct angle, and since the unit circle reaches only heights between $-1$ and $1$, so does the sine function.

For your other question, the answer is fairly simple. The unit circle can be defined as the set $$S=\{(x,y)| x^2+y^2 = 1\}$$
Using the definition of the sine and cosine functions, it is simple to show that this set is equal to the set $$\{(\cos \theta, \sin\theta)|\theta\in[0,2\pi)\}.$$
The circle centered around $0$ with a radius different than $1$, say the radius $R\neq 1$, is defined as $$S_R=\{(x,y)| x^2+y^2 = R^2\}$$
and can be shown to be equal to the set
$$\{(R\cos \theta, R\sin\theta)|\theta\in[0,2\pi)\}.$$
A: If the radius was not $1$ then you could not say that the  x and y-coordinates of a point on a circle are described  by $\cos \theta$ and $\sin \theta$, since $\cos^2 \theta +\sin^2 \theta =1$.
So you could say that the sine (or cosine) of real angles is always between negative one and positive one because the hypotenuse is the longest side of a right angled triangle.
