The unit circle and special angles

This unit circle is really an activity you should do, not a document for you to read passively.

Draw this unit circle as big as you can on a full sheet of paper. I strongly recommend that you draw it yourself rather than printing it out, as you will remember it better that way. You can think of it as a circle divided into twelfths superimposed on a circle divided into eighths, with vertical and horizontal lines going from each point on the circle to the x and y axes. Draw it very carefully - perhaps a compass, protractor and ruler will be helpful.

Now label the points around the outside of the circle by their corresponding radian measures. Thus the point at $(1,0)$ will be labelled $0$ and the point at $(0,1)$ will be labelled $\pi/2$, and so on. Do this carefully; every angle is either a fraction or a whole number times $\pi$, and if its a fraction the denominator is either $2$, $3$, $4$ or $6$. One way to do this is to first label all the angles which are multiples of $\pi/6$, and then reduce these fractions, and then do the same for all the angles which are multiples of $\pi/4$. (I.e. $0 \pi/6$, $\pi/6$, $2\pi/6$, $3\pi/6$, $4\pi/6$, $5\pi/6$, etc., and then reduce these to $0$, $\pi/6$, $\pi/3$, $\pi/2$, $2\pi/3$, etc.)

Now label all the points on the x and y axes. Label the origin with $0$, of course, and label the four points where the circle intersects the axes with plus or minus $1$. Now of the remaining points, there are four which, just by inspection, are clearly equal to plus or minus $1/2$. All the points on the axes coming from the division of the circle into eighths (the angles which are multiples of $\pi/4$) are equal to plus or minus $1/\sqrt{2}$. Then the remaining points are equal to plus or minus $\sqrt{3}/2$. Double check your work by seeing if it is consistent with the values you know (or can look up in the book) for sine and cosine of these special angles.

Another way to label the points on the axes comes from the following cool fact: Along each axis, in order from the origin out to the circle, the points you encounter are (ignoring sign) equal to $\sqrt{0}/2$, $\sqrt{1}/2$, $\sqrt{2}/2$, $\sqrt{3}/2$, $\sqrt{4}/2$. Of course most of those numbers can be written, and should be written, more simply (e.g. $\sqrt{4}/2 = 1$), but this shows that there is actually a pattern.

Now that you have your final, carefully drawn unit circle with all angles and all x and y axes labelled, you can easily look up, for example $\sin(7\pi/6)$. Just locate $7\pi/6$ on the circumference of the circle, remember that sine is the y-coordinate of that point, so follow the horizontal line from that point to the y-axis and read off the value there. Also, if you want to know $\sin^{-1}(-1/2)$, find $-1/2$ on the y-axis and go horizontally over to the point on the unit circle in the first or fourth quadrant. Here you have to convert the angle to a negative one; you might find it useful to label some negative angles as well, working clockwise from the angle $0$.

I hope that is useful.