WEBVTT

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Hello.

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Well, welcome to the next
presentation in the

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trigonometry modules.

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

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we've done so far.

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

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

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the sine, the cosine, and
the tangent functions.

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And the mnemonic we used to
memorize that was sohcahtoa.

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Let me write that down.

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Sohcahtoa.

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

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Let me draw a right triangle.

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This is a right angle here.

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This is the hypotenuse.

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Let me label the hypotenuse, h.

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

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use this angle right here.

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Theta, we'll call this theta.

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Whatever.

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

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And that's an o.

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So soh tells us that sine
is equal to opposite

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over hypotenuse.

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Cosine is equal to
adjacent over hypotenuse.

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And tangent is equal to
opposite over adjacent.

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

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

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nature and should make
a lot of sense to you.

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But this definition, using a
right triangle like this,

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actually breaks down
at certain points.

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Actually, at a lot of points.

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For example, what happens
as this angle right here

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approaches 90 degrees?

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

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triangle, can you?

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Then it would be like a
rectangle or something.

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But you could actually probably
figure out what happens as

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it approaches 90 degrees.

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But the definition, this
definition, breaks

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down for that.

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Also, what happens if
this angle is negative?

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

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Or what happens if
it's 800 degrees?

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Or you know, 8 pi radians?

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

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But obviously, this definition
starts to break down.

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Because we couldn't even
draw a right triangle that

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has those properties.

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

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of this definition.

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

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and the tangent functions to be
defined for angles greater than

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

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So let's draw a unit circle.

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

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circle of radius 1.

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And let's make-- let me see.

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

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OK.

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

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This is an angle, right?

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

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And this is a radius, right?

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And we said that this
has a radius 1.

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So the length of this
line is 1, right?

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Because it just goes
from the origin to the

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outside of the circle.

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So it has a radius of 1.

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

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Let me just drop a
line from here.

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So there I have a
right triangle.

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So if we use the old
definition we learned before.

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Let's just focus
on sine for now.

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So sine is equal to
opposite over hypotenuse.

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

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This is the right angle.

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So what's the opposite
angle of this?

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What's the opposite
side from this angle?

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I'm going to change to yellow.

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It's this side, right?

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This is the opposite side.

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And what's the hypotenuse?

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The hypotenuse is just
this radius, right?

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

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

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So what's the height of
this opposite side?

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Well, it's y, right?

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Because it's just the
height of that point.

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This is of height y.

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

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

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

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equal y over the hypotenuse.

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The hypotenuse is
this pink side here.

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And what's the length
of the hypotenuse?

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

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So it's 1.

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And y divided by 1?

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Well, that's just y.

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

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

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Well, we know that cosine
is equal to adjacent

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over hypotenuse.

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Well, what's the
adjacent side here?

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I'm running out of colors.

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The adjacent side is this
bottom side, right here.

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

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out of space, too.

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

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the adjacent side--
and what is that?

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What is this length?

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What is the length
of this side?

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Well, it's just x, right?

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

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

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that this distance is y.

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So this distance being
just x, we know that the

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adjacent length is x.

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

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And once again, the
hypotenuse is 1.

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So cosine of theta
is equal to x.

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I know what you're thinking.

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Sal, that's very nice and cute.

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

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But how is this really
different from what we

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were doing before?

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

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angle becomes 90 degrees,
now I can actually define

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what sine of theta is.

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Sine of theta now is just y.

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Is just the y-coordinate,
which is 1.

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

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

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or pi over 2 radians.

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This is pi over 2.

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This angle right here.

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And similar, cosine
of pi over 2 is 0.

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Because the x-coordinate
right here is 0.

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Let me do it with a
couple more examples.

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Oh, I'm forgetting the
tangent function.

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And you could probably
figure out now, what is the

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definition now we can use
for the tangent function?

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

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Because it's kind
of a normal angle.

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So in this green angle
here, tangent is

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opposite over adjacent.

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

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

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

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

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

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I'm getting confused
with terminology.

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It's where this line
intersects the circumference.

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

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The cosine of theta
is equal to x.

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

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

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little bit more sense to you.

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

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So that's my unit circle.

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And here's the coordinate axis.

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It's one of them.

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And here is the other one.

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

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Theta equals pi over 2.

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Well, pi over 2 is right here.

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

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And now, we just figure
out where it intersects

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the unit circle.

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

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has a radius of 1.

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

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where it intersects
the unit circle.

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So that's just 1.

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What's cosine of pi over 2?

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

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intersect the unit circle.

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And the x-coordinate here is 0.

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And what's the tangent
of pi over 2?

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This is interesting.

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The tangent of pi over 2.

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

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

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The y-coordinate is 1.

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So it equals 1/0.

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So this is undefined.

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

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define itself at
certain points.

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

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And you'll see that it
approaches infinity.

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

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theta equals pi, right?

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That's like 180 degrees.

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That's this point right here.

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So sine of pi.

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What's the y-coordinate
at this point?

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Well, this point is
negative 1 comma 0.

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So the y-coordinate is 0.

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What's the x-coordinate?

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Cosine of pi.

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That's negative 1.

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

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It's y/x.

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

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Hopefully this makes sense.

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

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

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

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See you soon.

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Bye.

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