The Unit Circle


You should know the sines, cosines and tangents that come from 30-60-90 right triangles and isosceles right triangles. Remember that $\pi/6$ radians is 30 degrees, $\pi/4$ radians is 45 degrees, and $\pi/3$ radians is 60 degrees.

DO:  Draw the right triangle with legs of length $1$ and $\sqrt 3$ and hypotenuse of length $2$.  The angle opposite the shorter leg (of length 1) is $\pi/6$ and the angle opposite the longer leg (of length $\sqrt 3$) is $\pi/3$.  If you memorize this triangle, you are set, since from this triangle you get all the following values:

$\sin(\pi/6) = \frac{1}{2};$ $\cos(\pi/6) = \frac{\sqrt{3}}{2};$ $\tan(\pi/6)= \frac{1}{\sqrt{3}}$
$\sin(\pi/4)=\frac{1}{\sqrt{2}};$ $\cos(\pi/4) = \frac{1}{\sqrt{2}};$ $\tan(\pi/4)=1$
$\sin(\pi/3)=\frac{\sqrt{3}}{2},$ $\cos(\pi/3)=\frac{1}{2};$ $\tan(\pi/3)=\sqrt{3}$

From these values, we can figure out the trig functions of common angles in the second, third and fourth quadrant.  It is easy to use symmetry to do this. These ideas and more are developed in the following two videos: