How to interpret trig functions in other quadrants?

Question

When working in Quadrant I, things make sense to me. Angles are bound between 0 and 90 degrees and we can talk about sin, cos, and tan in terms of ratios of sides of triangles.

But when we increase the angle and start moving into other quadrants I no longer understand how you're supposed to interpret things. I'll see someone draw a triangle in Quadrant III like normal but then somehow it's still reconciling with this huge angle that's sprawling all the way over from 0 on the unit circle.

How am I supposed to interpret these trigonometric functions in other quadrants?

Answer 1

Trigonometric functions have clear definition and geometrical meaning which work good for angles in all quadrants beyond the definition in terms of ratios of sides of triangles which can be viewed as an application.

Notably

• $\cos x$ and $\sin x$ are the coordinates of the point M on the trigonometric circle
• $\tan x$ is the y coordinate of the intersection between the vertical line from (1,0) and the line OM
• ...

and so on.