Trouble understanding sine, cosine of angles greater than $90$ degrees I recently learned about the trigonometric functions of angles greater than $90$ degrees, and I'm having a hard time understanding the concept.
We worked with a diagram like this:

My teacher proceeded to tell the class that the sine of angle $\theta$ was equal to the ratio between the opposite side ($y$) and the hypotenuse ($r$).
I don't quite understand this logic. I'm familiar with trigonometric functions, but as I remember them, they applied to angles inside a right-triangle, not an angle outside of it. This seems counter-intuitive to me.
Shouldn't the sine ratio (or cosine, for that matter) represent the angle $180 - \theta$ located inside the right triangle, rather than outside of it? An explanation would be greatly appreciated, as this is a key part of our current unit.
 A: When trigonometric functions like sine and cosine are applied to situations where we're dealing with angles that are greater than or equal to 90 degrees, the logic based purely on the right triangle definition of trigonometric functions as we know it breaks because in elementary trigonometry the sum of the angles in a right triangle (or any other triangle, for that matter) can't be greater than $180^\circ$. That's obvious.
Let's say you have a 135-degree angle ($90^\circ+45^\circ=135^\circ$), as in your picture. Basically, you still, kind of, use the right triangle idea, but you now must take into account the fact that the legs of the triangle can have negative values because the whole idea that the angle can be greater than or equal to 90 degrees comes from the unit circle which we work with in the Cartesian coordinate system. Therefore, when $x$ is negative, the length of the corresponding leg has a negative value. When $y$ is negative, the length of the corresponding leg is going to be negative as well. Does this make sense to you?
If you find my explanation confusing, here's a more-to-the-point version: forget about the right triangle definition for angles greater than or equal to 90 degrees. It doesn't really work. Now you work in the Cartesian coordinate system where the hypotenuse equals $1$, $\cos\theta$ is the x-coordinate and $\sin\theta$ is the y-coordinate.
