https://en.wikipedia.org/wiki/Sine#Series_definition

>>7779711
There are lots of different representations of sine, but the simplest would be the Taylor series: [eqn] \sin (x) = \sum^{ \infty } _ { n=0 } \frac { (-1)^{n} } { (2n + 1)! } x^{ 2n +1} [/eqn] You'd have to hold quite a few terms for it to converge to a reasonable accuracy as x gets larger.

>>7779711
The sine (and cosine) function, which you probably know can be described by the proportional sides of a right triangle, is a function which meets a specific set of properties (like sin^2(x) +cos^2(x) = 1, among many others) and we can prove that this is the ONLY function which meets these properties, proving it is unique. So you could argue that we calculate it through use of a right triangle and prove that it is a unique function in this respect in that it is the only function to have certain properties.

In addition, however, there are the complex forms as well.

Sin(x) = (e^(ix) - e^(-ix))/2i

This is also a method of calculating sine. Simply plug it into that definition.
Note that it is still a unique function despite having 2 representations because we can prove (through Euler's identity) that both representations are equal and thus they are the same function.

Understand?

https://en.wikipedia.org/wiki/Lookup_table#Computing_sines

>>7779711
sine is just a ratio; that graph is using radians (rather than degrees) aka pi means 180, 2pi means 360

so, as you see on the graph, sine of pi radians is equal to 0; sin(pi)=0 you can do the same thing for the rest of the graph

in order to actually understand it you have to look up the unit circle, that the sine ratio is using certain triangles in the unit circle

>>7779711
OP think about what you're asking.

An "equation" is a statement of equality, a statement which says one thing equals another thing. I think what you are asking is "what is the function that represents sine", and the answer is of course the sine function. The sine function has many equivalent definitions though. One is the Taylor Series definition. Another is the complex form definition. Another is the definition by right triangles.

What you really seem to be asking though is "what is the polynomial function which functions the same as the sine function?", and the answer to that is the complex form of the sine function, seen here: >>7779719 .
Keep in mind that, as one can prove, the sine function is a unique function, so every form of it (complex, Taylor Series, etc.) must be equal to every other form of it.

>>7779711

I was more or less under the impression that sine just represented a particular ratio determined by an angle. Not necessarily an equation.

>>7780291
yes but op wants to know how you find the given ratio for the given angle.

>>7779711
How's high school, anon?