The Unit Circle

Date: 4/2/96 at 23:54:55
From: Anonymous
Subject: Determining values of trig functions with the unit circle

Hello,
I am currently an AP Calculus student.  Last year, I and many of 
my classmates studied trig/analytic geometry under a foreign 
exchange teacher from Estonia.  The guy was absolutely fantastic 
and knew his math extremely well.  He did not, however, really 
stress the unit circle and how to use it to determine values of 
trig functions.  For example: sin (Pi/6) or cos (Pi/3)... I learned 
that sin 30 is 1/2 and that cos 60=sin (90-60)=sin 30=1/2 through 
repetition.  Now, however, we are studying integrals with inverse 
trig functions and application of and familiarity with the unit 
circle is imperative.  If possible, could you describe how to 
"read" the unit circle and perhaps post a copy of the circle 
itself?  I would appreciate your help.

     Thanks

     John Zeese

Date: 4/4/96 at 11:10:6
From: Doctor Aaron
Subject: Re: Determining values of trig functions with the unit 
circle

Hi,

I'll start by trying to clarify what sines and cosines are, and 
then I'll talk a little bit about the unit circle.  Most of what 
is happening with trig can be described with individual triangles.  
The circle is just a way to describe the collection of triangles 
and give meaning to the sign (+ or -) of trig functions.

The Sine.  If theta is an interior angle of a right triangle, then 
the sine of theta, written: sin(theta) is the ratio of the 
opposite side to the hypotenuse.

The Cosine.  If theta is an interior angle of a right triangle, 
then the cosine of theta, written cos(theta) is the ratio of the 
adjacent side to the hypotenuse.

One thing that you expressed confusion about is that if theta is 
between 0 and 90 (as it must be from the above definitions of sine 
and cosine)

Cos (90 - theta) = Sin (theta) and Sin (90-theta) = Cos (theta).

Think about a right triangle ABC
A          where a refers to the angle at vertex A, etc. 
|\         that is, b is a right angle.
| \                         BC
|  \       Well, Sin (a) = ---- because BC is the opposite side
|   \                       AC
|    \                                                BC
|     \     and AC is the hypotenuse, then Cos (c) = ---- because
|______\                                              AC
B      C    we moved to angle c so BC is now the adjacent side, and
            AC is still the hypotenuse.  These ratios are identical, 
so we can write Sin (a) = Cos (c).  Since the sum of the interior 
angles in any triangle is 180 degrees, and we already established 
that b is a 90 degree angle, then the angle measures of a and c 
must sum to 90.

Then we can rewrite c = 90-a or a = 90-c, and substituting into 
the above equality, we get Sin (a) = Cos (90-a) or Sin (90-c) = 
Cos (c).

Now I will address your primary question: What does all of this 
have to do with the unit circle.  I am limited to ascii graphics, 
so I'm not going to draw a circle here, but if you get a pencil 
and paper and draw a circle of radius 1, centered at the origin of 
the x-y plane.  Then the circle should intercept the x-axis at +1 
and -1, and also intercept the y-axis as +1 and -1.

Pick a point on the unit circle (we'll start with one in the first 
quadrant) connect it to the center with a radius and drop a 
vertical that connects it to the x-axis.  Note that this describes 
a triangle.  Also note that the length of the hypotenuse is 1.  
All we did to describe the triangle was to pick a point on the 
circle, so we can think of the unit circle as describing the 
collection of right triangles with hypotenuse 1.  We could 
generalize this to say that the circle of radius r describes the 
collection of all triangles with hypotenuse of length r, but any 
right triangle is similar to some right triangle with hypotenuse 
length 1, and 1 is such a nice number to have in the denominator, 
that we restrict ourselves to the unit circle.

What does this tell us?

One nice thing about a triangle determined by a point on the unit 
circle (X,Y), is that we can read the trig values of the angle 
formed by the radius and the x-axis, pretty easily.  The radius of 
the unit circle is the hypotenuse of the triangle, and the region 
[0,X] (or [X,0] if X<0) is one leg of the triangle, we can see 
that the length of this leg is X.  The other leg is the line that 
goes from (X,0) to (X,Y) so we can see that that length is Y.  
Then if theta is the angle less than 90 defined by the hypotenuse 
and the x-axis, 

              opposite       Y
Sin (theta) = ---------- = -------
              hypotenuse     1

and 
               adjacent      X
Cos (theta) = ----------- = ----
               hypotenuse    1

The last thing that I'm going to talk about is how we extend our 
notion of Sine and Cosine to angles that measure greater than 90 
degrees.  

Now pick any point on the unit circle (go for one not in the first 
quadrant) make the triangle, except this time, we're going to 
measure the angle that the radius makes with the positive x-axis 
measured counter-clockwise from the positive x-axis.  That is, if 
you choose (0,1) then your angle is 90 degrees, and if you choose 
(0,-1) your angle is 270 degrees.

The difference is that this time we are going to say that the 
length of the legs are X and Y even if X and/or Y are negative, 
then we will be able to get negative sines and cosines.

If you choose a point in the second quadrant X is negative but Y 
is positive so Cos (theta) = X will be negative, but Sin (theta) = 
Y will be positive.  Similary Both Cosine and Sine will be 
negative in the third quadrant and Cosine will be positive, but 
Sine will be negative in the fourth quadrant.

I hope that I have adequately addressed your question.  If not 
please write back or work through your difficulty with a teacher 
or friend.

-Doctor Aaron,  The Math Forum