Q:

Trigonometric ratios for angles greater than $90^\circ$?

The trigonometric ratios of an angle greater than $90^\circ$ are equal to the supplementary angle's ratios.
I'm just clarifying this, but the ratios don't actually exist for angles greater than $90^\circ$ right? (since by definition these ratios are of a right angle triangle). Is it just mathematical custom to assume the ratios of an angle greater than $90^\circ$ to be equivalent to its supplementary angle's ratios? For convenience?

A:

Instead of ratios in right triangles (which as you notice make sense only for acute angles), one can consider the cosine and sine defined as the $x$ and $y$ coordinate of a point that moves around a unit circle. This works for all angles -- and for acute angles you can inscribe a right triangle in the first quadrant of the unit circle and see that the unit-circle definition matches the right-triangle one.

After 90°, the cosine becomes negative, because the point is now to the left of the $y$ axis (so the $x$ coordinate is negative).
After 180°, the sine becomes negative too -- both coordinates of the moving point are now negative.

A:

There are two possible definitions of the trigonometric ratios:

The trigonometric ratios can be defined for angles greater than $0^\circ$ and less than $90^\circ$ using right triangles.  In particular, $\sin(\theta)$ is defined as the ratio of the lengths of the opposite leg and the hypotenuse, and $\cos(\theta)$ is defined as the ratio of the lengths of the adjacent leg and the hypotenuse.
The trigonometric ratios can be defined for any angle using the unit circle.  In this definition, $\sin(\theta)$ is the $y$-coordinate of a point on the unit circle with angle $\theta$, and $\cos(\theta)$ is the $x$-coordinate of a point on the unit circle with angle $\theta$.

The unit circle definition is the same as the triangle definition for angles between $0$ and $90^\circ$, but is more general since it works for any angle.  The following picture from Wikipedia illustrates this definition:

For each point, the $x$-coordinate is the cosine, and the $y$-coordinate is the sine.
This picture only shows angles between $0^\circ$ and $360^\circ$, but you can extend to less than $0^\circ$ by continuing clockwise around the circle, or to greater than $360^\circ$ by continuing counterclockwise.
The following pictures show graphs of $\sin(x)$ and $\cos(x)$ for $-2\pi\leq x\leq2\pi$.  (The $x$-axis is the angle measured in radians.)

