BAM!!! Mr. Tarrou. Ok. We got the basics down. I hope you understand from the first video.
Let's just get to this unit circle so that you can see what the final product is going
to look like. Now I keep saying unit circle. All that is, is a circle that is centered
on the origin. All of our angles are in standard position. The vertex is on the origin. And,
the radius is one. That is the big deal. The unit circle, the radius is one. So whoopee
do...right?! Now what that means is that I have got all of these trig functions here.
Sine is y over r. Cosine is x over y. And tangent which I am not going to put on the
unit circle as ahh... something that you have to memorize is y over x. And i also said that
the unit circle is mostly comprised of values that come off of these two triangles, the
60-30 triangle you learned you learned from geometry and the 45-45 triangle. Now, because
of this definition of the unit circle has a definition of 1, some really cool things
happens. Now you do want to definitely remember that Sine is y over r, but for a unit circle
only, the r value is always going to always be 1. Thats what the definition says. The
radius is 1. So, when you look at Sine as y over r, at least as far as the unit circle
is concerned, that's y over 1 or just the y values. So, the Sine is the y-coordinate
of all the points we're gonna have in a minute around the unit circle. For a unit circle
whose radius is 1, the Cosine is gonna be x over 1. So as you look at all these points
that we'll have done in a minute around the unit circle, your Cosine values are just simply
the x-coordinate... The x-coordinate on a circle whose radius is 1 and the center is
in the origin or on the origin of the coordinate plane: the x y coordinate plane. Now, I'm
also going to be referring to these two special triangles from geometry, but the hypotenuse
of a.... right triangle will act as the radius of the circle. So, we want these radius values,
these hypotenuse values, to be 1. So I'm gonna come through here and effectively just divide
all of these values by 2. So, if I divide the value 2 by 2 I get 1. And then if I divide
this length by 2 I get one half. And if I divide this length by 2, I get square root
3 over 2. I don't particularly prefer this ratio, or these... these sides for a 60-30
triangle, if I'm just using regular geometry and I'm just dealing with triangles that are
around... you know just floating around in space. But for purposes of setting up the
unit circle, I want the hypotenuse to be 1. And so i want this hypotenuse to also be 1.
I'm just changing the sides of these triangles. So I'm going to do that by dividing everything
by the square root of 2. AHEM... To get all these lengths or to get the hypotenuse - excuse
me - in to that length of 1 so I can place it as a ratio of my unit circle. So square
root of 2 divided by square root of 2 is 1. 1 divided by square root of 2, and when you
rationalize the denominator, is gonna be the value of square root of 2 over 2. And then
same thing because the two legs of a 45-45 are equal, it's also going to be the square
root of 2 over 2. Now, there's 16 units uh angles around the unit circle. I know I just
did this in an earlier video, so I'll go quickly with this. You count by 30... And count off
your... 12 of your 16 angles so... 30 degrees. 60 degrees. 90 degrees. 120. 150. 2... excuse
me 180. 210. 240. 270. 300 degrees. And then finally 330 degrees. Then for the other 4
units on the unit circle you wanna count by 45s. So 45. uh 135. 180. 225. 270. 315. And
then finally of course either zero or 360 degrees. Now, I went much slower in my original
video for setting up the unit circle. 180 divided by 30, you have to divide by 6. Now
180 is equal to pi radians, so this is gonna be equal to pi over 6, pi over 3... I'm counting
by one sixths but I wanna show you build up the coordinates not the value so I'm just
gonna give these out. One sixth, pi over 3, pi over 2, 2 pi over 3, 5 pi over 6, 7 pi
over 6. uh... 1, 2, 3, 4 pi over 3, 3 pi over 2, or 5 pi over 3. See, even I count these
out and I teach this for a living... And 11 pi over 6. And then you count by pi over 4.
So 1 fourth, pi over 2, 3 pi over 4, pi, 5 pi over 4, 3 pi over 2 or 6 pi over 4, and
then finally 7 pi over 4. Okay. Again, these degrees and these 16 degrees and thee 16 radian
measures must be memorized eventually to quicken your pace of your work.. to make your work
more efficient. And if you're one of my students, to pass all the unit circle quizzes I'm gonna
hit you - excuse me - hit you with. Okay now. Let's keep in mind that on the unit circle,
Cosine is x over r, yes. And you do wanna memorize that ratio of x over r. But on the
unit circle, the x coordinate - or excuse me - the radius is 1. So, to simplify that
on the unit circle, the Cosine is simply the x coordinate that we're gonna get around these
16 points. The y is Sine of y, Sine of Theta - excuse me - the Sine of Theta is y over
r. And on the unit circle the radius is always 1. So on the unit circle only, the Sine value
is simply gonna be the y coordinate of the point. So we have a circle whose radius is
equal to 1. So, just remember as we go around this. So now what's the coordinate at this
point at 0? That coordinate is 1, 0. So the Cosine of, well, Theta is your first number.
So the Cosine of 0 is gonna be 1 and the Sine of 0 is gonna be, again, the y coordinate.
Cosine is the x coordinate and the Sine is the y coordinate. So, at 0 the Cosine of 0
is 1 and the Sine of 0 is equal to 0. Now at 30 degrees. Let's go ahead and set up that
triangle... with 30 here. And we have square root of 3 over 2, one half, and 1. So the
Cosine of 30 is x over r so the the Cosine of 30 is square root of 3 over 2. And the
y value at 30 degrees - when you rotate 30 degrees - the y coordinate of that point,
or the Sine of 30 is y over r or just simply one half. Okay and I'll do a 45 degree angle
up here for 135. But let's just fill it in for now. 45 degrees... And you can look here.
At 45 degrees, the x and the y coordinate is square root of 2 over 2. And the same for
the y coordinate - square root of 2 over 2. At 60 degrees... Now, again, I wanna.. I'm
gonna do a triangle, a reference triangle, on each quadrant, but I can't keep drawing
over this one. You won't be able to read it. So, at 60 degrees, the Cosine value is going
to be... I kinda wanna just.. you know sort of like put my finger on it. 0 to 30, that's
60 degrees. And opposite of 60 degrees you have... or adjacent of 60 degrees you have
one half. So the Cosine of 60 is one half. And the y coordinate is square root of 3 over
2. It's kind of bugging me that I'm not drawing that. So, let's go ahead and do that. I feel
like you might need to see that even though I was gonna do it in another quadrant. There!
So, at 60 degrees, when i draw my reference triangle, with the 60 and the right angle.
Adjacent to 60 you have a length of 1/2, an x coordinate of one half, and a y coordinate
opposite of 60, you have that length of square root of 3 over 2. And again, for unit circle
purposes, the radius or the hypotenuse. I prefer to have that said 1. So the Cosine
of 60 is x over one or one half. And the Sine of 60 is square root of 3 over 2. And we see
that right there. And at 90 degrees, we can't set up a triangle because we're straight up
and down on the x axis... acts as one of the legs of the triangle so there's no way you
can draw a triangle there. But we know that if we're at the origin, we would simply go
up one length, one unit, that that coordinate is 0, 1. So the Cosine of 90 is 0. And the
Sine of 90, the y coordinate, we're up one unit, so the Sine value is 1. Now let's go
over here to 120 degrees. Well, I'm gonna draw a triangle at 135, so I'm not gonna draw
this one. But, at 120 degrees, the reference angle for 60 degrees is 60. In the previous
video, I had to find where the reference angle was. If I rotate over to 120 degrees, I'm
60 degrees off of the x axis. So these... this 120 and 60 degrees has the same 60 degree
reference angle. Let me just draw a tiny piece of that right there. Okay so, 120 and 60 degrees
you're gonna have very familiar looking coordinates. Only when you're in quadrant 2 the x's are
negative. So, this will be negative one half and square root of 3 over 2. Now i will draw
a proper triangle for 135 degrees. So let's do that. Pick a different color here... let's
say red... If we rotate over to 135 degrees... And there'd be an angle in the standard position
angle here of 135 degrees... Well, again, you can't make a right triangle out of an
obtuse angle but what you can do is make a right triangle as a reference triangle, right?
We wanna draw those against the x axis, not the y axis. And at 135 degrees, you're 45
degrees off of the x axis. So there's a 45 degree reference angle here. Well, look at
the sides we have. As you move to the left, your x's are negative. And as you go up, your
y values are positive. And then the radius is gonna be set 1. So the Cosine of 135 degrees
is x over r or negative square root of 2 over 2. And the y coordinate is also square root
of 2 over 2. You see how these coordinates are starting to match up? The only difference
is because I'm in quadrant two, my x coordinates are negative instead of positive like they
were in quadrant one. At 150 degrees where they're again 30 degrees off of the x axis,
this point will have the same coordinates as this point, except of course again, the
x is going to be negative. So negative square root of 3 over 2 and positive one half. At
pi, we're at negative 1,0. At 210, we're at negative square root of 3 over 2 and negative
one half. Now we're getting into a um, negative y value because of course we're moving down
- and I'm running out of time - se we have a 30 degree reference angle. So these points
are all gonna have something in common, except that my x's are negative and my y's are negative
and let me just give you the final answers here. Negative square root of 2 over 2, negative
square root of 2 over 2. Negative one half, negative square root of 3 of 2. WHERE DID
THE TIME GO?? 0, negative one. One half, negative square root of 2. You should be noticing that
these coordinates are starting to repeat because all of the points that have very similar coordinates,
except for a sign change, all have matching reference angles. So um, 30, 150, 210, and
330 all have a 30 degree reference angle so they all have very similar coordinates. It's
just with the sign change and that pairing matches up all the way around the unit circle.
I'm Mr. Tarrou! That's the unit circle! BAMM!!!! You will notice, in the 30 seconds i have
left, 45, 135, 225, and 315 again all have matching coordinates except for sign changes
because they all have the same reference angle of 45 degrees. 60, 120, 240, and 300 as well
have matching reference angles of 60 degrees and therefore they have the same coordinates
only with sign changes. Thank you very much for listening and letting me help! I am Mr.
Tarrou. Go do your homework!