Angles Beyond The Right Triangle

A right triangle can tell you a lot about an angle, but only when the angle is between $0^\circ$ and $90^\circ$. Imagine you measure the steepness of a ramp, so you draw a right triangle and label one acute angle. That works. Now imagine the ramp tips past straight up, or you keep rotating the ramp arm around a full circle. The triangle picture breaks, not because the angle stopped existing, but because there is no right triangle with an angle like $120^\circ$ or $210^\circ$ sitting inside it.

Here is the basic limitation. In a right triangle, one angle is $90^\circ$, and the other two must add to $90^\circ$. So each of those two angles has to be acute. That means the familiar triangle definitions of sine and cosine are trapped in the first quadrant.

If you have ever seen something like $\sin(150^\circ)$ and felt stuck, that feeling is reasonable. The question is not how to force a triangle to fit. The question is what picture of an angle still works when the angle goes past a right angle, or even goes negative.

Pick some angles, decide whether a right triangle view makes sense for each, and notice where it fails.