TITLE: What do sine, tan, cos actually mean?
QUESTION [26 upvotes]: I know that $\sin\theta=\frac{y}{r}$ and $\cos\theta=\frac{x}{r}$.
My question is: is $\sin$ a function of $\theta$, as in $\sin (\theta$)?
If yes, why is there no $\theta$ on the right hand side of the equation? For example, $f(x) = mx + c$. Here $f$ is a function of $x$.
If not, what is the relationship between the symbols $\sin$ and $\theta$?
Another question is: why can we write $\theta = \sin ^{-1} (\frac{y}{r})$ and what does it actually mean?

REPLY [4 votes]: The trigonometric functions relate an angle to a length. More precisely, in a rectangle triangle, the sine, cosine and tangent of an angle are the ratio of two of the sides.

$$\cos(\theta)=\frac xr,\ \sin(\theta)=\frac yr,\ \tan(\theta)=\frac yx.$$
For convenience, we will now assume that $r=1$, as this doesn't change the proportions.
$$\color{green}{\cos}(\theta)=\color{green}x,\ \color{blue}{\sin}(\theta)=\color{blue}y,\ \color{red}{\tan}(\theta)=\frac{\color{blue}y}{\color{green}x}.$$
As you vary $\theta$, the opposite vertex describes a circle, and $x$ and $y$ vary accordingly. So the trigonometric functions are indeed... functions. Here is their plot.

In the expression $$\sin(\theta)=y,$$
you don't see $\theta$ explicitly in the right-hand side, but it appears implicitly in $y$. This is a definition of the sine resulting from a "geometric computation". Other expressions exist, such as the Taylor's development which allows you to compute numerically
$$\sin(\theta)=\theta-\frac{\theta^3}{6}+\frac{\theta^5}{120}-\frac{\theta^7}{5040}+\frac{\theta^9}{362880}\cdots$$
where $\theta$ must be measured in radians (a full turn, $360°$, is $2\pi$ radians).
The sine function transforms an angle to a ratio of lengths. You can invert the relation and ask what is the angle that gives some ratio. This inverse function is called the arcsine, and is denoted $\arcsin$ or $\sin^{-1}$.
$$\sin(\theta)=y\longleftrightarrow\theta=\arcsin(y).$$
Similarly, you can evaluate with a Taylors' development
$$\arcsin(y)=y + \frac{y^3}{6} + \frac{3y^5}{40} + \frac{5y^7}{112} +\frac{35y^9}{1152}\cdots $$