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.
Best Answer
If you think of the graph of $\sin(x)$ it's a nice periodic function, the graph is a wave. It's very useful in physics (for example) to have functions that model wave behavior. For this you need to allow the angle to go for multiple cycles. Without such functions things like Fourier analysis would be impossible.