system
Given the following post, respond as if you're commenting on Reddit:
human
How are the sine and cosine functions derived?

I understand that for certain angles, using the relationships between the sides of 30-60-90 and 45-45-90 triangles can be used to find the value of sine and cosine.

But what I don't understand is how those functions are found for different angles, like 11 degrees. In practice you use a calculator, but the calculator's answer has to come from somewhere.
gpt
There are a few different ways, but one way is to look at the circle. Take a unit circle in the *x*-*y* plane (i.e. the graph of *x*^2 + *y*^2 = 1). Any point point *p* on the circle forms a right triangle: the three sides of the triangle are formed from the line from the origin to *p* (Line R), the perpendicular line from the *x*-axis to *p* (Line X), and the perpendicular line from the *y*-axis to *p* (Line Y). Define θ as the angle between the *x*-axis and Line R. Then, define cos(θ) as the length of Line X and define sin(θ) as the length of Line Y. This is one way of defining sine and cosine.