How does the unit circle work for trigonometric ratios of non-acute angles? How does the unit circle work for trigonometric ratios of obtuse angles? I know that the x coordinate is $cos (\theta )$ and the y coordinate is $sin (\theta )$. But I understand these in context of only acute angles? I don't understand why the unit circle definition works for other than acute angles? Somebody please provide me some good intuition. 
 A: The "magnitude" of $\cos \theta, \sin \theta$ of an obtuse angles can be measure in terms of the acute angle its terminal arm forms with the $x$-axis, positive or negative. So an angle of, say, $\theta = \frac 34 \pi$ has its terminal arm pointing in the direction of Quadrant II, and the absolute values of $\cos\theta$, $\;\sin \theta$ are the same as the values of $\theta' = \pi/4$, the angle formed by the terminal arm of $\theta$ with respect to the negative $x$-axis: $\theta' = \pi - \frac{3\pi}{4}$.
The sign of that magnitude can be determined by the quadrant in which the angle's terminating ray is directed. (Recall that $\cos \theta$ corresponds with the $x$ coordinate, $\sin \theta$ with $y$):
Quadrant I: $\cos \theta, \sin \theta \gt 0$
Quadrant II: $\cos \theta \lt 0$, $\;\sin\theta > 0$
Quadrant III: $\cos \theta \lt 0,\;\;\sin\theta \lt 0$.
Quadrant IV: $\cos \theta \gt 0,\;\;\sin\theta \lt 0$.
A: This intuition can be found in the history of trig. Looking into my history of math notes (a course mandatory for my degree) I learned that once upon a time there was a man Hipparchus. He is regarded as a ofundaing father of what's now known as trig. Apparently there was a desire at that time to have tables that would convert any point on a unit circle expressed in Polar form, into corresponding Cartesian form. Hipparchus made those tables and assigned "functions" to the idea how every angle (measured from the pos. x-axis) of a point P on the circle would "correspond" to (x,y) coordinates. That's how the modern cosine and sine could be defined. Right triangle geometry is then derived from here. In this realm, the functions cosine and sine are born from the desire to have a "translation" from Polar to Cartesian form. In this light, all quandrants of the unit circle are covered and thus the sign of cosine and sine are understood accordingly.
