Unit Circle Definition of Trig Functions

Using the unit circle to define the sine, cosine, and tangent functions

Hello.

Well, welcome to the next
presentation in the

trigonometry modules.

Just to start off a little
bit, let's review what

we've done so far.

In the last couple modules, we
learned the definitions-- or at

least, I guess, we could call
it a partial definition-- of

the sine, the cosine, and
the tangent functions.

And the mnemonic we used to
memorize that was sohcahtoa.

Let me write that down.

Sohcahtoa.

And what that told us is, let's
say we had a right triangle.

Let me draw a right triangle.

This is a right angle here.

This is the hypotenuse.

Let me label the hypotenuse, h.

Let me label this-- and so we
want to figure out, we want to

use this angle right here.

Theta, we'll call this theta.

Whatever.

Then this is the adjacent side,
and this is the opposite side.

And that's an o.

So soh tells us that sine
is equal to opposite

over hypotenuse.

Cosine is equal to
adjacent over hypotenuse.

And tangent is equal to
opposite over adjacent.

And I think, by this point--
and especially if you did some

of the exercises on the Khan
Academy-- that should be second

nature and should make
a lot of sense to you.

But this definition, using a
right triangle like this,

actually breaks down
at certain points.

Actually, at a lot of points.

For example, what happens
as this angle right here

approaches 90 degrees?

You can't have two 90
degree angles in a right

triangle, can you?

Then it would be like a
rectangle or something.

But you could actually probably
figure out what happens as

it approaches 90 degrees.

But the definition, this
definition, breaks

down for that.

Also, what happens if
this angle is negative?

Or what happens if this angle
is more than 90 degrees?

Or what happens if
it's 800 degrees?

Or you know, 8 pi radians?

Not that 800 and 8 pi
radians are the same thing.

But obviously, this definition
starts to break down.

Because we couldn't even
draw a right triangle that

has those properties.

So now I'm going to introduce
you to an extension

of this definition.

It's really the same thing, but
it allows the sine, the cosine,

and the tangent functions to be
defined for angles greater than

or equal to pi over 2, or 90
degrees, or less than 0.

So let's draw a unit circle.

So this is just the coordinate
axis, and here is a

circle of radius 1.

And let's make-- let me see.

Let me make sure I'm using
the correct pen tool.

OK.

So let's call this right
here-- so this is theta.

This is an angle, right?

Between the x-axis and this
line I just drew here.

And this is a radius, right?

And we said that this
has a radius 1.

So the length of this
line is 1, right?

Because it just goes
from the origin to the

outside of the circle.

So it has a radius of 1.

And now I'm going to draw
a right triangle again.

Let me just drop a
line from here.

So there I have a
right triangle.

So if we use the old
definition we learned before.

Let's just focus
on sine for now.

So sine is equal to
opposite over hypotenuse.

Let's apply that to this
right triangle right here.

This is the right angle.

So what's the opposite
angle of this?

What's the opposite
side from this angle?

I'm going to change to yellow.

It's this side, right?

This is the opposite side.

And what's the hypotenuse?

The hypotenuse is just
this radius, right?

And let's just say that this
point, where it intersects

the circle-- let's call this
point right here x comma y.

So what's the height of
this opposite side?

Well, it's y, right?

Because it's just the
height of that point.

This is of height y.

So sine of this angle right
here, sine of theta, is

going to equal the opposite
side-- which is this yellow

side, which is just the
y-coordinate-- is going to

equal y over the hypotenuse.

The hypotenuse is
this pink side here.

And what's the length
of the hypotenuse?

Well, it's the radius
of this unit circle.

So it's 1.

And y divided by 1?

Well, that's just y.

So we see that sine of
theta is equal to y.

Let's do the same thing
for cosine of theta.

Well, we know that cosine
is equal to adjacent

over hypotenuse.

Well, what's the
adjacent side here?

I'm running out of colors.

The adjacent side is this
bottom side, right here.

So that would equal-- so
if I said-- I'm running

out of space, too.

Cosine of theta would equal
this gray side-- which is

the adjacent side--
and what is that?

What is this length?

What is the length
of this side?

Well, it's just x, right?

If this is the point x, y then
this distance here is x and we

already learned this distance--
or we already observed--

that this distance is y.

So this distance being
just x, we know that the

adjacent length is x.

So we say cosine of theta is
equal to x over the hypotenuse.

And once again, the
hypotenuse is 1.

So cosine of theta
is equal to x.

I know what you're thinking.

Sal, that's very nice and cute.

Cosine of theta equals x,
sine of theta equals y.

But how is this really
different from what we

were doing before?

Well, if I define it this way,
now all of a sudden when the

angle becomes 90 degrees,
now I can actually define

what sine of theta is.

Sine of theta now is just y.

Is just the y-coordinate,
which is 1.

If theta is equal to-- I'm
going to make sure it's very

messy right here-- if theta
is equal to 90 degrees,

or pi over 2 radians.

This is pi over 2.

This angle right here.

And similar, cosine
of pi over 2 is 0.

Because the x-coordinate
right here is 0.

Let me do it with a
couple more examples.

Oh, I'm forgetting the
tangent function.

And you could probably
figure out now, what is the

definition now we can use
for the tangent function?

Well, going back-- let's
use this green theta here.

Because it's kind
of a normal angle.

So in this green angle
here, tangent is

opposite over adjacent.

So tangent now, we
can define as y/x.

And remember, these y's and x's
that we're using are the point

on the unit circle where the
angle that's defined by this--

by whatever-- where the radius
that is subtended by this

angle, or I guess the arc,
intersects-- actually,

I'm getting confused
with terminology.

It's where this line
intersects the circumference.

The coordinate of that-- the
sine of theta is equal to y.

The cosine of theta
is equal to x.

And the tangent of
theta is equal to y/x.

Let's do a couple of examples
and hopefully this'll make a

little bit more sense to you.

Let me try to really fast
draw a new unit circle.

So that's my unit circle.

And here's the coordinate axis.

It's one of them.

And here is the other one.

So if we use the angle-- let's
use the angle pi over 2, right?

Theta equals pi over 2.

Well, pi over 2 is right here.

It's a 90 degree angle, if
you wanted to use degrees.

And now, we just figure
out where it intersects

the unit circle.

And once again, this is
a unit circle, so it

has a radius of 1.

So we can see that sine of pi
over 2 equals the y-coordinate

where it intersects
the unit circle.

So that's just 1.

What's cosine of pi over 2?

Well, it's just the
x-coordinate, where you

intersect the unit circle.

And the x-coordinate here is 0.

And what's the tangent
of pi over 2?

This is interesting.

The tangent of pi over 2.

Well, the tangent we
defined now as y/x.

So the y-coordinate, this
is the point 0, 1, right?

The y-coordinate is 1.

So it equals 1/0.

So this is undefined.

So still, we don't have a
tangent function that can

define itself at
certain points.

But in the next module, we're
actually going to graph this.

And you'll see that it
approaches infinity.

And similarly, we could try to
find the functions for when

theta equals pi, right?

That's like 180 degrees.

That's this point right here.

So sine of pi.

What's the y-coordinate
at this point?

Well, this point is
negative 1 comma 0.

So the y-coordinate is 0.

What's the x-coordinate?

Cosine of pi.

That's negative 1.

And of course, what's the
tangent of pi radians?

It's y/x.

So it's 0 over negative
1, which equals 0.

Hopefully this makes sense.

Now in the next module I'll
actually graph these points.

And you'll see how it all comes
together and why it is useful

to define the sine, the cosine,
the tangent functions this way.

See you soon.

Bye.