Why did we ever need to define the trig functions of angles greater than 90 degrees or less than 0 degrees? What is the use of applying trig functions to such angles?
If we apply the trig functions on a regular right triangle, it makes sense. We can get the ratio of two sides and find out an unknown side if there is a known side (and the other way around).
Let's say that I have a right triangle in which an angle x is 30 degrees and the hypotenuse is 20 cm. I have to find the length of side AY, which is opposite to angle x . Well I can use the function sin(30 degrees), which comes out to be 1/2 . Now 1/2 = AY / 20 . And after solving it we get AY = 10.
Or let's say that I have a right triangle in which I have to find an angle x. The side opposite to x is 10cm and the hypotenuse is 20 cm. Then 10/20 = 1/2. What is the arcsin of 1/2? 30 degrees. Angle x is 30 degrees.
But what use is it to take the sine of an angle 120 degrees of an obtuse triangle? We are not getting a ratio of the sides or anything if we apply it to a non-right triangle.