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?

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.

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.)

You have the following relationships, where two have a sign change:

• $\sin(180^\circ - x^\circ) = \sin(x^\circ) $
• $\cos(180^\circ - x^\circ) = -\cos(x^\circ) $
• $\tan(180^\circ - x^\circ) = -\tan(x^\circ) $

One way to think about angles greater than $90^\circ$ is with Cartesian co-ordinates with $x$ and $y$ axes (so some of $x$ and $y$ values can be negative, but the radius or hypotenuse to the origin is non-negative) and the angle is measured anti-clockwise from the $x$ axis.