Why do the ratios of sine and cosine depend upon the angle being a part of a right triangle?

Answer 1

Mark is correct.

However, as you progress in mathematics, you will come to find other ways to define cos and sin which do NOT depend on a right triangle. Nevertheless, it is almost always helpful to remember the relationship between the trig functions and a right triangle.

Answer 2

Both are a ratio of one of the legs divided by a hypotenuse.