We're now going to study the unit circle a little bit more,
and see how it extends, I guess we could say, the traditional
SOH-CAH-TOA definitions of functions.
And how we can actually use it to solve for angles that the
SOH-CAH-TOA definition of the trig functions actually
doesn't help us with.
So let's just, as a review, remember what
SOH-CAH-TOA told us.
I'll write it up here in this corner.
I don't want to get confusing, because I don't want to
write over too much.
SOH, CAH, TOA.
I'm sorry.
I got it all jumbled up here.
That told us that if we have a right angle that the sine of an
angle in the right angle is equal to the opposite side
over the hypotenuse.
The cosine of an angle is equal to the adjacent
side over hypotenuse.
And the tangent side is equal to the opposite
over the adjacent side.
And this worked fine for us.
But if you think about it, what happens when that angle
approaches 90 degrees?
Because you can't have two 90 degree angles in a right angle.
Or what happens when that angle is greater than 90 degrees?
Or what if it went negative?
And that's why, if you remember from the previous video,
we needed the unit circle definition.
So let's review the unit circle definition.
Let me erase this.
Dum di-dum di-dum.
I actually got this unit circle-- I think I got
it from Wikipedia.
But I want to give due credit to whoever I did
it get for, this drawing of the unit circle.
But the unit circle kind of extends that
SOH-CAH-TOA definition.
It tells us if we have a unit circle-- and this is a picture
of a unit circle here-- a unit circle is just a circle
centered at the origin, centered at the point 0, 0,
and it has a radius of 1.
So it intersects the x-axis at 1, 0 and negative 1, 0.
It intersects the y-axis at 0, 1 and 0, negative 1.
If we have a unit circle we define-- let's start with the
cosine of theta-- we define the cosine of theta as let's take
an angle that's between two radiuses in this unit circle.
And one radius is going to be the positive x-axis
between 0 and 1.
So one radius is going to be this line here.
And then we have the angle is the angle between you can kind
of view that as the base radius and some other radius.
So let's say this case right here.
And this would be our angle.
The unit circle definition tells us that the cosine of
this angle is equal to the x-coordinate where this radius
intersects the unit circle, and that the sine of this function
is equal to the y-coordinate where this point intersects
the unit circle.
So for example, in this case-- if you can read behind
my line-- this says 30 degrees equals pi/6.
So this angle right here is 30 degrees, or pi/6 radians.
And what this definition tells us is that the sine of 30
degrees is 1/2, and that the cosine of 30 degrees is
square root of 3/2.
And what I want to show you is that this unit circle
definition actually coincides with our SOH-CAH-TOA
definition, but then it extends it.
So let's see how we can get from that SOH-CAH-TOA
definition to this unit circle definition, and why they're
actually consistent with each other.
So let me erase some of what I have written here.
Let me get the eraser tool.
I'm going to erase what I had.
So let me go back to the pen tool, make it small again.
OK.
I think I'm all set.
So let's go back to that theta.
Let's say that this is the theta.
And as we said, this angle is 30 degrees or pi/6.
Let's drop a line from that point to the x-axis.
And as we see this line is perpendicular, so this
is a 90 degree angle.
And if this is a 30 degree angle here-- this is 30.
Right?
Theta equals 30.
This is 30, this is 90.
What is this angle?
Well, this is a 60 degree angle, because they
add up to 180.
So this is a 30-60-90 triangle.
Interesting.
And what do you remember about 30-60-90 triangles?
Well, the side opposite the 30 degree side is 1/2 the
length of the hypotenuse.
I hope you remember that.
I don't want to confuse you too much.
So this is the side opposite the 30 degree side.
Right?
And what's the hypotenuse?
This is the hypotenuse.
And what's the length of this hypotenuse?
Well it's 1, because this is a unit circle and this is the
radius of the unit circle.
So the length of this hypotenuse is 1, and so the
length of this side, which is opposite the 30 degree
angle, is going to be 1/2.
Right?
And I'm just using the 30-60-90 triangles that we've
done previous videos on.
And what's the side opposite the 60 degree side?
Well once again it's square root of 3/2
times the hypotenuse.
And so it's square root of 3/2.
Right?
So we can figure out that this side is square root of 3/2,
and that this side is 1/2.
So a couple of things we can figure out.
Just by looking at this we can immediately say, well what's
the coordinate of this point?
Well it's x-coordinate is right here.
Right?
It's x-coordinate would be square root of 3/2.
That's this right here.
This distance.
And it's y-coordinate would be the length of this side of
the right triangle, or 1/2.
And there we have it right here.
It was already written for us.
The x-coordinate is the square root of 3/2 and
the y-coordinate is 1/2.
And now what I want to show you is why this x-coordinate can be
taken as the cosine of theta, and why this y-coordinate can
be taken as the sine of theta.
Well what does SOH-CAH-TOA tell us?
Well let's start with the cosine.
So SOH, CAH, TOA.
So CAH.
Cosine is adjacent over hypotenuse, right?
Well, in this triangle I just drew, what is the adjacent
side to this angle?
Right?
Because we're trying to figure out the cosine of this
angle, this 30 degrees.
Well the adjacent side to this angle is, of course, this side.
Right?
So adjacent is square root of 3/2.
We figured that out just now.
And what's the hypotenuse?
Well the hypotenuse is this side, which has length 1
because it was the unit circle and that's the radius of it.
So the cosine of this angle using the SOH-CAH-TOA
definition is square root of 3-- the adjacent side--
over the hypotenuse 1.
So square root of 3/2 over 1, which is the square root
of 3/2, which was the same thing as the x-coordinate.
Similarly, we can look at SOH.
Sine equals opposite over hypotenuse.
Well what's the opposite side?
It's 1/2.
And the hypotenuse is 1 here.
So the sine is just 1/2 over 1.
And so we have it here.
So that's why the unit circle definition isn't kind of a
replacing definition for the SOH-CAH-TOA definition, it's
really just an extension that allows us-- I mean, for 30
degrees SOH-CAH-TOA worked, for 45 degrees SOH-CAH-TOA worked,
for 60 degrees it would work.
But once you get to 90 it becomes a little bit more
difficult if you use traditional SOH-CAH-TOA and you
try to draw a right triangle that has two 90 degree angles
in it-- because you couldn't.
And especially once you get to angles that are larger than 90
or angles that actually could even go negative.
It's not drawn here in the unit circle, but 330 degrees is the
same thing as negative 30 degrees, because you could go
either way in the circle.
And you could keep going around the circle.
You could figure out the sine or the cosine of, you know, 1
million degrees if you just keep going around the circle.
So hopefully this gives you a sense of the unit circle
definition of the sine and cosine functions.
And, of course, the tangent function is always just the
sine over the cosine, or the y over the x.
And you could use the unit circle definition
for that as well.
And I'll leave it for you as an exercise to try to derive all
of these other values using this unit circle, and using
what you already know about 30-60-90 triangles and what you
already know about 45-45-90 triangles, or what you
already know about the Pythagorean theorem.
And you should be able to figure out all of these values
going around the unit circle.
Hopefully that was helpful.
See you soon.