[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:01.28,Default,,0000,0000,0000,,
Dialogue: 0,0:00:01.28,0:00:02.21,Default,,0000,0000,0000,,Hello.
Dialogue: 0,0:00:02.21,0:00:06.15,Default,,0000,0000,0000,,Well, welcome to the next\Npresentation in the
Dialogue: 0,0:00:06.15,0:00:08.26,Default,,0000,0000,0000,,trigonometry modules.
Dialogue: 0,0:00:08.26,0:00:10.34,Default,,0000,0000,0000,,Just to start off a little\Nbit, let's review what
Dialogue: 0,0:00:10.34,0:00:13.07,Default,,0000,0000,0000,,we've done so far.
Dialogue: 0,0:00:13.07,0:00:15.32,Default,,0000,0000,0000,,In the last couple modules, we\Nlearned the definitions-- or at
Dialogue: 0,0:00:15.32,0:00:17.72,Default,,0000,0000,0000,,least, I guess, we could call\Nit a partial definition-- of
Dialogue: 0,0:00:17.72,0:00:19.88,Default,,0000,0000,0000,,the sine, the cosine, and\Nthe tangent functions.
Dialogue: 0,0:00:19.88,0:00:24.25,Default,,0000,0000,0000,,And the mnemonic we used to\Nmemorize that was sohcahtoa.
Dialogue: 0,0:00:24.25,0:00:26.19,Default,,0000,0000,0000,,Let me write that down.
Dialogue: 0,0:00:26.19,0:00:26.62,Default,,0000,0000,0000,,Sohcahtoa.
Dialogue: 0,0:00:26.62,0:00:32.49,Default,,0000,0000,0000,,
Dialogue: 0,0:00:32.49,0:00:35.32,Default,,0000,0000,0000,,And what that told us is, let's\Nsay we had a right triangle.
Dialogue: 0,0:00:35.32,0:00:36.70,Default,,0000,0000,0000,,Let me draw a right triangle.
Dialogue: 0,0:00:36.70,0:00:39.72,Default,,0000,0000,0000,,
Dialogue: 0,0:00:39.72,0:00:40.85,Default,,0000,0000,0000,,This is a right angle here.
Dialogue: 0,0:00:40.85,0:00:42.36,Default,,0000,0000,0000,,This is the hypotenuse.
Dialogue: 0,0:00:42.36,0:00:44.68,Default,,0000,0000,0000,,Let me label the hypotenuse, h.
Dialogue: 0,0:00:44.68,0:00:49.57,Default,,0000,0000,0000,,Let me label this-- and so we\Nwant to figure out, we want to
Dialogue: 0,0:00:49.57,0:00:50.73,Default,,0000,0000,0000,,use this angle right here.
Dialogue: 0,0:00:50.73,0:00:52.11,Default,,0000,0000,0000,,Theta, we'll call this theta.
Dialogue: 0,0:00:52.11,0:00:52.78,Default,,0000,0000,0000,,Whatever.
Dialogue: 0,0:00:52.78,0:00:57.65,Default,,0000,0000,0000,,Then this is the adjacent side,\Nand this is the opposite side.
Dialogue: 0,0:00:57.65,0:00:58.77,Default,,0000,0000,0000,,And that's an o.
Dialogue: 0,0:00:58.77,0:01:02.31,Default,,0000,0000,0000,,So soh tells us that sine\Nis equal to opposite
Dialogue: 0,0:01:02.31,0:01:03.88,Default,,0000,0000,0000,,over hypotenuse.
Dialogue: 0,0:01:03.88,0:01:07.60,Default,,0000,0000,0000,,Cosine is equal to\Nadjacent over hypotenuse.
Dialogue: 0,0:01:07.60,0:01:11.25,Default,,0000,0000,0000,,And tangent is equal to\Nopposite over adjacent.
Dialogue: 0,0:01:11.25,0:01:14.31,Default,,0000,0000,0000,,And I think, by this point--\Nand especially if you did some
Dialogue: 0,0:01:14.31,0:01:17.59,Default,,0000,0000,0000,,of the exercises on the Khan\NAcademy-- that should be second
Dialogue: 0,0:01:17.59,0:01:19.47,Default,,0000,0000,0000,,nature and should make\Na lot of sense to you.
Dialogue: 0,0:01:19.47,0:01:23.46,Default,,0000,0000,0000,,But this definition, using a\Nright triangle like this,
Dialogue: 0,0:01:23.46,0:01:25.14,Default,,0000,0000,0000,,actually breaks down\Nat certain points.
Dialogue: 0,0:01:25.14,0:01:26.39,Default,,0000,0000,0000,,Actually, at a lot of points.
Dialogue: 0,0:01:26.39,0:01:29.64,Default,,0000,0000,0000,,For example, what happens\Nas this angle right here
Dialogue: 0,0:01:29.64,0:01:31.16,Default,,0000,0000,0000,,approaches 90 degrees?
Dialogue: 0,0:01:31.16,0:01:34.28,Default,,0000,0000,0000,,You can't have two 90\Ndegree angles in a right
Dialogue: 0,0:01:34.28,0:01:35.33,Default,,0000,0000,0000,,triangle, can you?
Dialogue: 0,0:01:35.33,0:01:37.62,Default,,0000,0000,0000,,Then it would be like a\Nrectangle or something.
Dialogue: 0,0:01:37.62,0:01:39.34,Default,,0000,0000,0000,,But you could actually probably\Nfigure out what happens as
Dialogue: 0,0:01:39.34,0:01:40.51,Default,,0000,0000,0000,,it approaches 90 degrees.
Dialogue: 0,0:01:40.51,0:01:42.98,Default,,0000,0000,0000,,But the definition, this\Ndefinition, breaks
Dialogue: 0,0:01:42.98,0:01:44.34,Default,,0000,0000,0000,,down for that.
Dialogue: 0,0:01:44.34,0:01:46.28,Default,,0000,0000,0000,,Also, what happens if\Nthis angle is negative?
Dialogue: 0,0:01:46.28,0:01:49.66,Default,,0000,0000,0000,,Or what happens if this angle\Nis more than 90 degrees?
Dialogue: 0,0:01:49.66,0:01:52.60,Default,,0000,0000,0000,,Or what happens if\Nit's 800 degrees?
Dialogue: 0,0:01:52.60,0:01:55.60,Default,,0000,0000,0000,,Or you know, 8 pi radians?
Dialogue: 0,0:01:55.60,0:01:57.83,Default,,0000,0000,0000,,Not that 800 and 8 pi\Nradians are the same thing.
Dialogue: 0,0:01:57.83,0:02:00.18,Default,,0000,0000,0000,,But obviously, this definition\Nstarts to break down.
Dialogue: 0,0:02:00.18,0:02:02.10,Default,,0000,0000,0000,,Because we couldn't even\Ndraw a right triangle that
Dialogue: 0,0:02:02.10,0:02:03.89,Default,,0000,0000,0000,,has those properties.
Dialogue: 0,0:02:03.89,0:02:07.63,Default,,0000,0000,0000,,So now I'm going to introduce\Nyou to an extension
Dialogue: 0,0:02:07.63,0:02:08.50,Default,,0000,0000,0000,,of this definition.
Dialogue: 0,0:02:08.50,0:02:12.92,Default,,0000,0000,0000,,It's really the same thing, but\Nit allows the sine, the cosine,
Dialogue: 0,0:02:12.92,0:02:16.97,Default,,0000,0000,0000,,and the tangent functions to be\Ndefined for angles greater than
Dialogue: 0,0:02:16.97,0:02:21.53,Default,,0000,0000,0000,,or equal to pi over 2, or 90\Ndegrees, or less than 0.
Dialogue: 0,0:02:21.53,0:02:22.90,Default,,0000,0000,0000,,So let's draw a unit circle.
Dialogue: 0,0:02:22.90,0:02:25.29,Default,,0000,0000,0000,,So this is just the coordinate\Naxis, and here is a
Dialogue: 0,0:02:25.29,0:02:27.70,Default,,0000,0000,0000,,circle of radius 1.
Dialogue: 0,0:02:27.70,0:02:30.54,Default,,0000,0000,0000,,And let's make-- let me see.
Dialogue: 0,0:02:30.54,0:02:32.92,Default,,0000,0000,0000,,Let me make sure I'm using\Nthe correct pen tool.
Dialogue: 0,0:02:32.92,0:02:33.67,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:02:33.67,0:02:39.93,Default,,0000,0000,0000,,So let's call this right\Nhere-- so this is theta.
Dialogue: 0,0:02:39.93,0:02:42.22,Default,,0000,0000,0000,,This is an angle, right?
Dialogue: 0,0:02:42.22,0:02:45.64,Default,,0000,0000,0000,,Between the x-axis and this\Nline I just drew here.
Dialogue: 0,0:02:45.64,0:02:47.01,Default,,0000,0000,0000,,And this is a radius, right?
Dialogue: 0,0:02:47.01,0:02:49.59,Default,,0000,0000,0000,,And we said that this\Nhas a radius 1.
Dialogue: 0,0:02:49.59,0:02:51.60,Default,,0000,0000,0000,,So the length of this\Nline is 1, right?
Dialogue: 0,0:02:51.60,0:02:54.02,Default,,0000,0000,0000,,Because it just goes\Nfrom the origin to the
Dialogue: 0,0:02:54.02,0:02:54.82,Default,,0000,0000,0000,,outside of the circle.
Dialogue: 0,0:02:54.82,0:02:56.94,Default,,0000,0000,0000,,So it has a radius of 1.
Dialogue: 0,0:02:56.94,0:02:59.89,Default,,0000,0000,0000,,And now I'm going to draw\Na right triangle again.
Dialogue: 0,0:02:59.89,0:03:02.50,Default,,0000,0000,0000,,Let me just drop a\Nline from here.
Dialogue: 0,0:03:02.50,0:03:05.92,Default,,0000,0000,0000,,So there I have a\Nright triangle.
Dialogue: 0,0:03:05.92,0:03:07.95,Default,,0000,0000,0000,,So if we use the old\Ndefinition we learned before.
Dialogue: 0,0:03:07.95,0:03:10.01,Default,,0000,0000,0000,,Let's just focus\Non sine for now.
Dialogue: 0,0:03:10.01,0:03:17.92,Default,,0000,0000,0000,,So sine is equal to\Nopposite over hypotenuse.
Dialogue: 0,0:03:17.92,0:03:20.28,Default,,0000,0000,0000,,Let's apply that to this\Nright triangle right here.
Dialogue: 0,0:03:20.28,0:03:22.03,Default,,0000,0000,0000,,This is the right angle.
Dialogue: 0,0:03:22.03,0:03:24.17,Default,,0000,0000,0000,,So what's the opposite\Nangle of this?
Dialogue: 0,0:03:24.17,0:03:25.84,Default,,0000,0000,0000,,What's the opposite\Nside from this angle?
Dialogue: 0,0:03:25.84,0:03:29.20,Default,,0000,0000,0000,,
Dialogue: 0,0:03:29.20,0:03:32.09,Default,,0000,0000,0000,,I'm going to change to yellow.
Dialogue: 0,0:03:32.09,0:03:32.96,Default,,0000,0000,0000,,It's this side, right?
Dialogue: 0,0:03:32.96,0:03:34.63,Default,,0000,0000,0000,,This is the opposite side.
Dialogue: 0,0:03:34.63,0:03:37.04,Default,,0000,0000,0000,,And what's the hypotenuse?
Dialogue: 0,0:03:37.04,0:03:42.50,Default,,0000,0000,0000,,The hypotenuse is just\Nthis radius, right?
Dialogue: 0,0:03:42.50,0:03:45.44,Default,,0000,0000,0000,,And let's just say that this\Npoint, where it intersects
Dialogue: 0,0:03:45.44,0:03:54.55,Default,,0000,0000,0000,,the circle-- let's call this\Npoint right here x comma y.
Dialogue: 0,0:03:54.55,0:04:00.45,Default,,0000,0000,0000,,So what's the height of\Nthis opposite side?
Dialogue: 0,0:04:00.45,0:04:01.30,Default,,0000,0000,0000,,Well, it's y, right?
Dialogue: 0,0:04:01.30,0:04:05.13,Default,,0000,0000,0000,,Because it's just the\Nheight of that point.
Dialogue: 0,0:04:05.13,0:04:06.57,Default,,0000,0000,0000,,This is of height y.
Dialogue: 0,0:04:06.57,0:04:13.17,Default,,0000,0000,0000,,So sine of this angle right\Nhere, sine of theta, is
Dialogue: 0,0:04:13.17,0:04:15.40,Default,,0000,0000,0000,,going to equal the opposite\Nside-- which is this yellow
Dialogue: 0,0:04:15.40,0:04:18.34,Default,,0000,0000,0000,,side, which is just the\Ny-coordinate-- is going to
Dialogue: 0,0:04:18.34,0:04:21.49,Default,,0000,0000,0000,,equal y over the hypotenuse.
Dialogue: 0,0:04:21.49,0:04:23.93,Default,,0000,0000,0000,,The hypotenuse is\Nthis pink side here.
Dialogue: 0,0:04:23.93,0:04:25.35,Default,,0000,0000,0000,,And what's the length\Nof the hypotenuse?
Dialogue: 0,0:04:25.35,0:04:27.75,Default,,0000,0000,0000,,Well, it's the radius\Nof this unit circle.
Dialogue: 0,0:04:27.75,0:04:29.60,Default,,0000,0000,0000,,So it's 1.
Dialogue: 0,0:04:29.60,0:04:31.10,Default,,0000,0000,0000,,And y divided by 1?
Dialogue: 0,0:04:31.10,0:04:31.86,Default,,0000,0000,0000,,Well, that's just y.
Dialogue: 0,0:04:31.86,0:04:40.04,Default,,0000,0000,0000,,So we see that sine of\Ntheta is equal to y.
Dialogue: 0,0:04:40.04,0:04:42.52,Default,,0000,0000,0000,,Let's do the same thing\Nfor cosine of theta.
Dialogue: 0,0:04:42.52,0:04:49.70,Default,,0000,0000,0000,,Well, we know that cosine\Nis equal to adjacent
Dialogue: 0,0:04:49.70,0:04:51.43,Default,,0000,0000,0000,,over hypotenuse.
Dialogue: 0,0:04:51.43,0:04:55.41,Default,,0000,0000,0000,,Well, what's the\Nadjacent side here?
Dialogue: 0,0:04:55.41,0:04:56.47,Default,,0000,0000,0000,,I'm running out of colors.
Dialogue: 0,0:04:56.47,0:05:00.20,Default,,0000,0000,0000,,The adjacent side is this\Nbottom side, right here.
Dialogue: 0,0:05:00.20,0:05:02.39,Default,,0000,0000,0000,,So that would equal-- so\Nif I said-- I'm running
Dialogue: 0,0:05:02.39,0:05:03.76,Default,,0000,0000,0000,,out of space, too.
Dialogue: 0,0:05:03.76,0:05:09.46,Default,,0000,0000,0000,,Cosine of theta would equal\Nthis gray side-- which is
Dialogue: 0,0:05:09.46,0:05:11.25,Default,,0000,0000,0000,,the adjacent side--\Nand what is that?
Dialogue: 0,0:05:11.25,0:05:14.35,Default,,0000,0000,0000,,What is this length?
Dialogue: 0,0:05:14.35,0:05:16.61,Default,,0000,0000,0000,,What is the length\Nof this side?
Dialogue: 0,0:05:16.61,0:05:18.43,Default,,0000,0000,0000,,Well, it's just x, right?
Dialogue: 0,0:05:18.43,0:05:23.07,Default,,0000,0000,0000,,If this is the point x, y then\Nthis distance here is x and we
Dialogue: 0,0:05:23.07,0:05:25.25,Default,,0000,0000,0000,,already learned this distance--\Nor we already observed--
Dialogue: 0,0:05:25.25,0:05:26.74,Default,,0000,0000,0000,,that this distance is y.
Dialogue: 0,0:05:26.74,0:05:29.10,Default,,0000,0000,0000,,So this distance being\Njust x, we know that the
Dialogue: 0,0:05:29.10,0:05:31.28,Default,,0000,0000,0000,,adjacent length is x.
Dialogue: 0,0:05:31.28,0:05:34.98,Default,,0000,0000,0000,,So we say cosine of theta is\Nequal to x over the hypotenuse.
Dialogue: 0,0:05:34.98,0:05:37.36,Default,,0000,0000,0000,,And once again, the\Nhypotenuse is 1.
Dialogue: 0,0:05:37.36,0:05:43.33,Default,,0000,0000,0000,,So cosine of theta\Nis equal to x.
Dialogue: 0,0:05:43.33,0:05:44.12,Default,,0000,0000,0000,,I know what you're thinking.
Dialogue: 0,0:05:44.12,0:05:45.82,Default,,0000,0000,0000,,Sal, that's very nice and cute.
Dialogue: 0,0:05:45.82,0:05:48.28,Default,,0000,0000,0000,,Cosine of theta equals x,\Nsine of theta equals y.
Dialogue: 0,0:05:48.28,0:05:49.76,Default,,0000,0000,0000,,But how is this really\Ndifferent from what we
Dialogue: 0,0:05:49.76,0:05:51.67,Default,,0000,0000,0000,,were doing before?
Dialogue: 0,0:05:51.67,0:05:56.16,Default,,0000,0000,0000,,Well, if I define it this way,\Nnow all of a sudden when the
Dialogue: 0,0:05:56.16,0:06:02.94,Default,,0000,0000,0000,,angle becomes 90 degrees,\Nnow I can actually define
Dialogue: 0,0:06:02.94,0:06:04.98,Default,,0000,0000,0000,,what sine of theta is.
Dialogue: 0,0:06:04.98,0:06:07.54,Default,,0000,0000,0000,,Sine of theta now is just y.
Dialogue: 0,0:06:07.54,0:06:10.19,Default,,0000,0000,0000,,Is just the y-coordinate,\Nwhich is 1.
Dialogue: 0,0:06:10.19,0:06:13.40,Default,,0000,0000,0000,,If theta is equal to-- I'm\Ngoing to make sure it's very
Dialogue: 0,0:06:13.40,0:06:15.61,Default,,0000,0000,0000,,messy right here-- if theta\Nis equal to 90 degrees,
Dialogue: 0,0:06:15.61,0:06:18.00,Default,,0000,0000,0000,,or pi over 2 radians.
Dialogue: 0,0:06:18.00,0:06:20.68,Default,,0000,0000,0000,,This is pi over 2.
Dialogue: 0,0:06:20.68,0:06:22.87,Default,,0000,0000,0000,,This angle right here.
Dialogue: 0,0:06:22.87,0:06:26.59,Default,,0000,0000,0000,,And similar, cosine\Nof pi over 2 is 0.
Dialogue: 0,0:06:26.59,0:06:31.65,Default,,0000,0000,0000,,Because the x-coordinate\Nright here is 0.
Dialogue: 0,0:06:31.65,0:06:33.06,Default,,0000,0000,0000,,Let me do it with a\Ncouple more examples.
Dialogue: 0,0:06:33.06,0:06:35.05,Default,,0000,0000,0000,,Oh, I'm forgetting the\Ntangent function.
Dialogue: 0,0:06:35.05,0:06:36.74,Default,,0000,0000,0000,,And you could probably\Nfigure out now, what is the
Dialogue: 0,0:06:36.74,0:06:39.65,Default,,0000,0000,0000,,definition now we can use\Nfor the tangent function?
Dialogue: 0,0:06:39.65,0:06:41.60,Default,,0000,0000,0000,,Well, going back-- let's\Nuse this green theta here.
Dialogue: 0,0:06:41.60,0:06:43.49,Default,,0000,0000,0000,,Because it's kind\Nof a normal angle.
Dialogue: 0,0:06:43.49,0:06:45.83,Default,,0000,0000,0000,,So in this green angle\Nhere, tangent is
Dialogue: 0,0:06:45.83,0:06:47.68,Default,,0000,0000,0000,,opposite over adjacent.
Dialogue: 0,0:06:47.68,0:06:54.04,Default,,0000,0000,0000,,So tangent now, we\Ncan define as y/x.
Dialogue: 0,0:06:54.04,0:06:57.76,Default,,0000,0000,0000,,And remember, these y's and x's\Nthat we're using are the point
Dialogue: 0,0:06:57.76,0:07:01.78,Default,,0000,0000,0000,,on the unit circle where the\Nangle that's defined by this--
Dialogue: 0,0:07:01.78,0:07:06.44,Default,,0000,0000,0000,,by whatever-- where the radius\Nthat is subtended by this
Dialogue: 0,0:07:06.44,0:07:09.15,Default,,0000,0000,0000,,angle, or I guess the arc,\Nintersects-- actually,
Dialogue: 0,0:07:09.15,0:07:10.94,Default,,0000,0000,0000,,I'm getting confused\Nwith terminology.
Dialogue: 0,0:07:10.94,0:07:13.81,Default,,0000,0000,0000,,It's where this line\Nintersects the circumference.
Dialogue: 0,0:07:13.81,0:07:17.80,Default,,0000,0000,0000,,The coordinate of that-- the\Nsine of theta is equal to y.
Dialogue: 0,0:07:17.80,0:07:19.75,Default,,0000,0000,0000,,The cosine of theta\Nis equal to x.
Dialogue: 0,0:07:19.75,0:07:22.74,Default,,0000,0000,0000,,And the tangent of\Ntheta is equal to y/x.
Dialogue: 0,0:07:22.74,0:07:27.02,Default,,0000,0000,0000,,Let's do a couple of examples\Nand hopefully this'll make a
Dialogue: 0,0:07:27.02,0:07:28.11,Default,,0000,0000,0000,,little bit more sense to you.
Dialogue: 0,0:07:28.11,0:07:31.55,Default,,0000,0000,0000,,
Dialogue: 0,0:07:31.55,0:07:35.25,Default,,0000,0000,0000,,Let me try to really fast\Ndraw a new unit circle.
Dialogue: 0,0:07:35.25,0:07:37.78,Default,,0000,0000,0000,,
Dialogue: 0,0:07:37.78,0:07:39.03,Default,,0000,0000,0000,,So that's my unit circle.
Dialogue: 0,0:07:39.03,0:07:42.81,Default,,0000,0000,0000,,
Dialogue: 0,0:07:42.81,0:07:44.96,Default,,0000,0000,0000,,And here's the coordinate axis.
Dialogue: 0,0:07:44.96,0:07:46.53,Default,,0000,0000,0000,,It's one of them.
Dialogue: 0,0:07:46.53,0:07:48.00,Default,,0000,0000,0000,,And here is the other one.
Dialogue: 0,0:07:48.00,0:07:58.18,Default,,0000,0000,0000,,
Dialogue: 0,0:07:58.18,0:08:04.01,Default,,0000,0000,0000,,So if we use the angle-- let's\Nuse the angle pi over 2, right?
Dialogue: 0,0:08:04.01,0:08:06.45,Default,,0000,0000,0000,,Theta equals pi over 2.
Dialogue: 0,0:08:06.45,0:08:09.67,Default,,0000,0000,0000,,Well, pi over 2 is right here.
Dialogue: 0,0:08:09.67,0:08:12.86,Default,,0000,0000,0000,,It's a 90 degree angle, if\Nyou wanted to use degrees.
Dialogue: 0,0:08:12.86,0:08:15.30,Default,,0000,0000,0000,,And now, we just figure\Nout where it intersects
Dialogue: 0,0:08:15.30,0:08:15.84,Default,,0000,0000,0000,,the unit circle.
Dialogue: 0,0:08:15.84,0:08:17.72,Default,,0000,0000,0000,,And once again, this is\Na unit circle, so it
Dialogue: 0,0:08:17.72,0:08:20.30,Default,,0000,0000,0000,,has a radius of 1.
Dialogue: 0,0:08:20.30,0:08:29.21,Default,,0000,0000,0000,,So we can see that sine of pi\Nover 2 equals the y-coordinate
Dialogue: 0,0:08:29.21,0:08:32.08,Default,,0000,0000,0000,,where it intersects\Nthe unit circle.
Dialogue: 0,0:08:32.08,0:08:34.63,Default,,0000,0000,0000,,So that's just 1.
Dialogue: 0,0:08:34.63,0:08:36.09,Default,,0000,0000,0000,,What's cosine of pi over 2?
Dialogue: 0,0:08:36.09,0:08:39.34,Default,,0000,0000,0000,,
Dialogue: 0,0:08:39.34,0:08:41.09,Default,,0000,0000,0000,,Well, it's just the\Nx-coordinate, where you
Dialogue: 0,0:08:41.09,0:08:42.22,Default,,0000,0000,0000,,intersect the unit circle.
Dialogue: 0,0:08:42.22,0:08:46.06,Default,,0000,0000,0000,,And the x-coordinate here is 0.
Dialogue: 0,0:08:46.06,0:08:47.90,Default,,0000,0000,0000,,And what's the tangent\Nof pi over 2?
Dialogue: 0,0:08:47.90,0:08:49.70,Default,,0000,0000,0000,,This is interesting.
Dialogue: 0,0:08:49.70,0:08:52.77,Default,,0000,0000,0000,,The tangent of pi over 2.
Dialogue: 0,0:08:52.77,0:08:55.45,Default,,0000,0000,0000,,Well, the tangent we\Ndefined now as y/x.
Dialogue: 0,0:08:55.45,0:08:58.91,Default,,0000,0000,0000,,So the y-coordinate, this\Nis the point 0, 1, right?
Dialogue: 0,0:08:58.91,0:09:01.61,Default,,0000,0000,0000,,The y-coordinate is 1.
Dialogue: 0,0:09:01.61,0:09:02.46,Default,,0000,0000,0000,,So it equals 1/0.
Dialogue: 0,0:09:02.46,0:09:05.40,Default,,0000,0000,0000,,
Dialogue: 0,0:09:05.40,0:09:06.75,Default,,0000,0000,0000,,So this is undefined.
Dialogue: 0,0:09:06.75,0:09:09.80,Default,,0000,0000,0000,,So still, we don't have a\Ntangent function that can
Dialogue: 0,0:09:09.80,0:09:11.64,Default,,0000,0000,0000,,define itself at\Ncertain points.
Dialogue: 0,0:09:11.64,0:09:14.66,Default,,0000,0000,0000,,But in the next module, we're\Nactually going to graph this.
Dialogue: 0,0:09:14.66,0:09:17.47,Default,,0000,0000,0000,,And you'll see that it\Napproaches infinity.
Dialogue: 0,0:09:17.47,0:09:21.58,Default,,0000,0000,0000,,And similarly, we could try to\Nfind the functions for when
Dialogue: 0,0:09:21.58,0:09:24.10,Default,,0000,0000,0000,,theta equals pi, right?
Dialogue: 0,0:09:24.10,0:09:27.06,Default,,0000,0000,0000,,That's like 180 degrees.
Dialogue: 0,0:09:27.06,0:09:28.57,Default,,0000,0000,0000,,That's this point right here.
Dialogue: 0,0:09:28.57,0:09:31.45,Default,,0000,0000,0000,,So sine of pi.
Dialogue: 0,0:09:31.45,0:09:33.82,Default,,0000,0000,0000,,What's the y-coordinate\Nat this point?
Dialogue: 0,0:09:33.82,0:09:37.72,Default,,0000,0000,0000,,Well, this point is\Nnegative 1 comma 0.
Dialogue: 0,0:09:37.72,0:09:39.88,Default,,0000,0000,0000,,So the y-coordinate is 0.
Dialogue: 0,0:09:39.88,0:09:40.94,Default,,0000,0000,0000,,What's the x-coordinate?
Dialogue: 0,0:09:40.94,0:09:43.52,Default,,0000,0000,0000,,Cosine of pi.
Dialogue: 0,0:09:43.52,0:09:45.26,Default,,0000,0000,0000,,That's negative 1.
Dialogue: 0,0:09:45.26,0:09:49.57,Default,,0000,0000,0000,,And of course, what's the\Ntangent of pi radians?
Dialogue: 0,0:09:49.57,0:09:51.22,Default,,0000,0000,0000,,It's y/x.
Dialogue: 0,0:09:51.22,0:09:54.40,Default,,0000,0000,0000,,So it's 0 over negative\N1, which equals 0.
Dialogue: 0,0:09:54.40,0:09:55.76,Default,,0000,0000,0000,,Hopefully this makes sense.
Dialogue: 0,0:09:55.76,0:09:58.73,Default,,0000,0000,0000,,Now in the next module I'll\Nactually graph these points.
Dialogue: 0,0:09:58.73,0:10:04.45,Default,,0000,0000,0000,,And you'll see how it all comes\Ntogether and why it is useful
Dialogue: 0,0:10:04.45,0:10:09.01,Default,,0000,0000,0000,,to define the sine, the cosine,\Nthe tangent functions this way.
Dialogue: 0,0:10:09.01,0:10:10.16,Default,,0000,0000,0000,,See you soon.
Dialogue: 0,0:10:10.16,0:10:11.46,Default,,0000,0000,0000,,Bye.
Dialogue: 0,0:10:11.46,0:10:12.00,Default,,0000,0000,0000,,