# How do we define trigonometric functions?

1. Jul 29, 2014

### Mr Davis 97

I'm having a problem understanding exactly why trig functions are defined the way they are. Of course, the definition in terms of 0 to 90 degree angles within right triangles is easy: the functions just give the ratio of the sides given the angle. However, I don't understand how or why trig functions are defined for angles greater than 90 degrees. How does this relate to right angles? Why is it useful to define the trig functions as the points on the unit circle? That definition seems arbitrary, and not useful. Once we go beyond 90 degrees, why is the subject even called trigonometry, if it mostly only relates to the points on the unit circle? Why did we decide to define trig functions this way after defining them as ratios of the sides of right triangles? I hope that somebody can answer these question to put my mind at ease, because as of now, this doesn't make much sense to me.

2. Jul 29, 2014

### gopher_p

I made this post in a related topic a while ago. I think it's relevant here.

3. Jul 29, 2014

### Mr Davis 97

Thank you, that helps a lot. Knowing that they are not fundamentally the same thing helps me understand why we define them the way we do. But following this, I have another question. What compelled us to define functions in terms of the unit circle? I see how the right-triangle defined functions have a lot of practical applicability, but I don't see the reason that went on to define trig functions in the way of the unit circle in the first place.

4. Jul 30, 2014

### bhillyard

Technically it doesn't have to be a unit circle. But if it isn't the resulting definitions will require the radius of the circle to be considered. The reason for considering a definition in terms of a circle is that once you are considering angles greater than 90o then we want a definition that, for angles less than 90, is equivalent to the definitions in terms of right angled triangles.