Hey it’s Professor Dave, let’s memorize the unit circle.
Math students often have trouble with this thing.
It’s called the unit circle, and it’s just a circle with a radius of one superimposed
on the coordinate plane, with the center at the origin.
Then we have all the angles that are multiples of pi over six, as well as the multiples of
pi over four, and we have to know the sine and cosine values for these angles.
Let’s quickly go over some key points about the unit circle that will make it very easy
to remember.
First we must recall what sine and cosine are.
From SOHCATOA, sine means the length of the leg opposite the angle over the length of
the hypotenuse, and cosine means the length of the leg adjacent to the angle over the
length of the hypotenuse.
For each angle on this unit circle, we are constructing a triangle by drawing a line
from this point, straight down to the X axis.
The hypotenuse of this triangle is the radius of the circle, which is always one, and the
two legs are each some value less than one, depending on the angle.
What we must understand is that every point on this unit circle has an X coordinate equal
to cosine theta and a Y coordinate equal to sine theta.
Take zero as an example.
There’s no triangle, but since the radius of the circle is one, we know that the cosine
of zero must be one, because the X coordinate of this point is one.
Likewise, we know that the sine of zero must be zero, because the Y coordinate of this
point is zero.
It will be just as easy to figure out the values for half pi, as this point has the
coordinates zero, one.
So the cosine of half pi must be zero, while the sine of half pi must be one.
At pi, we get negative one and zero, and at three-halves pi we get zero and negative one.
Once we make it to two pi, we are back where we started, and the cycle continues.
Now what’s important to see, is that the cosine and sine of any angle other than these
must be somewhere in between negative one and one.
Let’s take pi over six, for example.
In the last clip we evaluated the trig functions for these two special triangles, so if you
missed that, go check it out now.
Otherwise, just take my word for it that the sine of pi over six radians, or thirty degrees,
is one half, and the cosine is root three over two.
That means that the coordinates of this point are root three over two, one half.
For quarter pi, we have this other special triangle, and both the sine and cosine of
forty-five are root two over two, so these are the coordinates of this point.
For two pi over six, or pi over three, we have the same values as for pi over six but
reversed, since it’s the same triangle, just looking at the other angle.
That means that these are the coordinates of this point.
And that wraps up the first quadrant.
Now here’s the thing that makes this so easy to remember.
We see some fractions and some roots, which may seem unrelated.
Just for the sake of creating a pattern, let’s replace zero with root zero over two, since
root zero is zero.
Let’s change one half into root one over two, since root one is one.
Then we have root two over two, root three over two, and let’s change one into root
four over two, since root four is two, and two over two is one.
Now, if we look at the sine values for these angles, which correspond to the Y coordinates
of the points, we can start at the X axis, where the Y value is root zero over two, we
climb up to root one over two, then root two over two, then root three over two, then root
four over two.
We could do the same thing with the X values, which correspond to cosine values for these
angles, starting with the Y axis, and the moving to the right in the positive direction.
Again we hit the same values.
So all we have to do is remember this one numerical sequence, and that the coordinates
of each point take the form of cosine theta, sine theta, and we will never forget any of
these values.
Let’s return these values to their simplified forms, and then extend things into quadrant two.
We can see that sine values, or Y coordinates, will remain positive, as these points are
above the X axis.
Five sixths pi will have a sine of one half just like one sixth pi.
Three quarters pi will have a sine of root two over two just like quarter pi.
Two thirds pi and one third pi each have a sine of root three over two.
Now for the cosine values, it’s the same thing, except that they are now negative,
as they are to the left of the Y axis.
Going into the third quadrant, both sine and cosine become negative, as X and Y are both
negative, but it’s still all the same numbers.
And into the fourth quadrant, cosines return to being positive, while sines remain negative.
Knowing this, we could take any of these angles, and immediately know the sine and cosine simply
by visualizing the unit circle.
Take four thirds pi for example.
To get the sine of this angle, we simply locate the angle, which can also be thought as this
reference angle, or one third pi, but in quadrant three.
Then, we recall that sine corresponds to the Y coordinate of this point, and we just move
downwards until we get there.
Zero, negative one half, negative root two over two, negative root three over two.
To get the cosine value, we just remember that cosine corresponds to the X coordinate
of the the point.
Zero, negative one half.
We aren’t calculating anything, we just have to memorize the locations of the angles,
this one numerical sequence, and what the coordinates of the points represent.
Once this is ingrained in you, you’ll never forget it, so just stare at this until it
make sense.
When you have it, it becomes trivial to evaluate trig functions for any common angle.
Take something like the tangent of fourteen pi over three.
Twelve thirds pi is the same as four pi, and that’s twice all the way around, so we can
subtract that from the angle to get two thirds pi.
That gets us right here.
Now tangent is sine over cosine, so let’s take the Y value of this point and divide
by the X value of this point.
And that gives us negative root three.
What about the cosecant of negative seventeen fourths pi?
This time we are going in the negative direction.
Sixteen fourths pi is the same as four pi, so if we get rid of that, we are just left
with negative a quarter pi.
That puts us here, where cosine is root two over two, and sine is negative root two over
two.
Cosecant is one over sine, so we take one over negative root two over two.
We can therefore just flip this over to get negative two over root two, and then we multiply
by root two over root two to get rid of the radical on the bottom.
That leaves us with negative root two.
It may take a little while to get accustomed to using the unit circle so quickly, but it’s
just a matter of practice.
When you’re ready, let’s check comprehension.