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## The unit circle definition of sine, cosine, and tangent

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# Unit circle

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## Video transcript

What I have attempted to
draw here is a unit circle. And the fact I'm
calling it a unit circle means it has a radius of 1. So this length from
the center-- and I centered it at the origin--
this length, from the center to any point on the
circle, is of length 1. So what would this coordinate
be right over there, right where it intersects
along the x-axis? Well, x would be
1, y would be 0. What would this
coordinate be up here? Well, we've gone 1
above the origin, but we haven't moved to
the left or the right. So our x value is 0. Our y value is 1. What about back here? Well, here our x value is -1. We've moved 1 to the left. And we haven't moved up or
down, so our y value is 0. And what about down here? Well, we've gone a unit
down, or 1 below the origin. But we haven't moved
in the xy direction. So our x is 0, and
our y is negative 1. Now, with that out of the way,
I'm going to draw an angle. And the way I'm going
to draw this angle-- I'm going to define a
convention for positive angles. I'm going to say a
positive angle-- well, the initial side
of the angle we're always going to do along
the positive x-axis. So you can kind of view
it as the starting side, the initial side of an angle. And then to draw a positive
angle, the terminal side, we're going to move in a
counterclockwise direction. So positive angle means
we're going counterclockwise. And this is just the
convention I'm going to use, and it's also the convention
that is typically used. And so you can imagine
a negative angle would move in a
clockwise direction. So let me draw a positive angle. So a positive angle might
look something like this. This is the initial side. And then from that, I go in
a counterclockwise direction until I measure out the angle. And then this is
the terminal side. So this is a
positive angle theta. And what I want to do is
think about this point of intersection
between the terminal side of this angle
and my unit circle. And let's just say it has
the coordinates a comma b. The x value where
it intersects is a. The y value where
it intersects is b. And the whole point
of what I'm doing here is I'm going to see how
this unit circle might be able to help us extend our
traditional definitions of trig functions. And so what I want
to do is I want to make this theta part
of a right triangle. So to make it part
of a right triangle, let me drop an altitude
right over here. And let me make it clear that
this is a 90-degree angle. So this theta is part
of this right triangle. So let's see what
we can figure out about the sides of
this right triangle. So the first question
I have to ask you is, what is the
length of the hypotenuse of this right triangle that
I have just constructed? Well, this hypotenuse is just
a radius of a unit circle. The unit circle
has a radius of 1. So the hypotenuse has length 1. Now, what is the length of
this blue side right over here? You could view this as the
opposite side to the angle. Well, this height is
the exact same thing as the y-coordinate of
this point of intersection. So this height right over here
is going to be equal to b. The y-coordinate
right over here is b. This height is equal to b. Now, exact same logic--
what is the length of this base going to be? The base just of
the right triangle? Well, this is going
to be the x-coordinate of this point of intersection. If you were to drop
this down, this is the point x is equal to a. Or this whole length between the
origin and that is of length a. Now that we have
set that up, what is the cosine-- let me
use the same green-- what is the cosine of my angle going
to be in terms of a's and b's and any other numbers
that might show up? Well, to think
about that, we just need our soh cah toa definition. That's the only one we have now. We are actually in the process
of extending it-- soh cah toa definition of trig functions. And the cah part is what
helps us with cosine. It tells us that the
cosine of an angle is equal to the length
of the adjacent side over the hypotenuse. So what's this going to be? The length of the
adjacent side-- for this angle, the
adjacent side has length a. So it's going to be
equal to a over-- what's the length of the hypotenuse? Well, that's just 1. So the cosine of theta
is just equal to a. Let me write this down again. So the cosine of theta
is just equal to a. It's equal to the x-coordinate
of where this terminal side of the angle
intersected the unit circle. Now let's think about
the sine of theta. And I'm going to do it in-- let
me see-- I'll do it in orange. So what's the sine
of theta going to be? Well, we just have to look at
the soh part of our soh cah toa definition. It tells us that sine is
opposite over hypotenuse. Well, the opposite
side here has length b. And the hypotenuse has length 1. So our sine of
theta is equal to b. So an interesting
thing-- this coordinate, this point where our
terminal side of our angle intersected the
unit circle, that point a, b-- we could
also view this as a is the same thing
as cosine of theta. And b is the same
thing as sine of theta. Well, that's interesting. We just used our soh
cah toa definition. Now, can we in some way use
this to extend soh cah toa? Because soh cah
toa has a problem. It works out fine if our angle
is greater than 0 degrees, if we're dealing with
degrees, and if it's less than 90 degrees. We can always make it
part of a right triangle. But soh cah toa
starts to break down as our angle is either 0 or
maybe even becomes negative, or as our angle is
90 degrees or more. You can't have a right triangle
with two 90-degree angles in it. It starts to break down. Let me make this clear. So sure, this is
a right triangle, so the angle is pretty large. I can make the angle even
larger and still have a right triangle. Even larger-- but I can never
get quite to 90 degrees. At 90 degrees, it's
not clear that I have a right triangle any more. It all seems to break down. And especially the
case, what happens when I go beyond 90 degrees. So let's see if we can
use what we said up here. Let's set up a new definition
of our trig functions which is really an
extension of soh cah toa and is consistent
with soh cah toa. Instead of defining cosine as
if I have a right triangle, and saying, OK, it's the
adjacent over the hypotenuse. Sine is the opposite
over the hypotenuse. Tangent is opposite
over adjacent. Why don't I just
say, for any angle, I can draw it in the unit circle
using this convention that I just set up? And let's just say that
the cosine of our angle is equal to the x-coordinate
where we intersect, where the terminal
side of our angle intersects the unit circle. And why don't we
define sine of theta to be equal to the
y-coordinate where the terminal side of the angle
intersects the unit circle? So essentially, for
any angle, this point is going to define cosine
of theta and sine of theta. And so what would be a
reasonable definition for tangent of theta? Well, tangent of theta--
even with soh cah toa-- could be defined
as sine of theta over cosine of theta,
which in this case is just going to be the
y-coordinate where we intersect the unit circle over
the x-coordinate. In the next few videos,
I'll show some examples where we use the unit
circle definition to start evaluating some trig ratios.