The unit circle definition of trigonometric function

Using the unit circle to extend the SOH CAH TOA definition of the basic trigonometric functions.

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 don't know how --
SOH CAH TOA

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.

And that told us --
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 [side]

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 [triangle].

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,

why we needed the
unit circle definition.

So, let's review the unit
circle definition.

Let me erase this.

Dum di-dum di-dum.

Let me erase --

I actually got this unit
circle [graphic],

I think I got it from Wikipedia.

But I want to give due
credit to whomever I did

get it [from], 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 [-1, 0].

It intersects the y-axis at
[0, 1] and [0, -1].

If we have a unit circle, we define --

Let's just say, let's start with
the cosine of theta (θ) --

We define the
cosine of theta (θ) 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 [take] 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 (π/6).

So this angle right here is
30 degrees, or pi/6 (π/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 [the] 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 (θ).

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.