[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:01.93,0:00:02.29,Default,,0000,0000,0000,,Voiceover:On the right-hand side we have
Dialogue: 0,0:00:02.29,0:00:04.32,Default,,0000,0000,0000,,a bunch of expressions\Nthat are just ratios
Dialogue: 0,0:00:04.32,0:00:07.51,Default,,0000,0000,0000,,of different information\Ngiven in these two diagrams.
Dialogue: 0,0:00:07.51,0:00:09.83,Default,,0000,0000,0000,,Then over here on the left we have
Dialogue: 0,0:00:09.83,0:00:11.86,Default,,0000,0000,0000,,the sine taken of angle MKJ,
Dialogue: 0,0:00:11.86,0:00:15.20,Default,,0000,0000,0000,,cosine of angle MKJ, and\Ntangent of angle MKJ.
Dialogue: 0,0:00:15.20,0:00:19.41,Default,,0000,0000,0000,,Angle MKJ is this angle right over here
Dialogue: 0,0:00:19.41,0:00:22.50,Default,,0000,0000,0000,,same thing as theta, so these two angles.
Dialogue: 0,0:00:22.50,0:00:25.19,Default,,0000,0000,0000,,These two angles have the same measure.
Dialogue: 0,0:00:25.19,0:00:26.77,Default,,0000,0000,0000,,We see that right over there.
Dialogue: 0,0:00:26.77,0:00:28.18,Default,,0000,0000,0000,,What we want to do is figure out
Dialogue: 0,0:00:28.18,0:00:30.55,Default,,0000,0000,0000,,which of these expressions are equivalent
Dialogue: 0,0:00:30.55,0:00:34.44,Default,,0000,0000,0000,,to which of these\Nexpressions right over here.
Dialogue: 0,0:00:34.44,0:00:36.71,Default,,0000,0000,0000,,I encourage you to pause the video
Dialogue: 0,0:00:36.71,0:00:40.31,Default,,0000,0000,0000,,and try to work this through on your own.
Dialogue: 0,0:00:40.31,0:00:42.90,Default,,0000,0000,0000,,Assuming you've had a go at it,
Dialogue: 0,0:00:42.90,0:00:44.34,Default,,0000,0000,0000,,let's try to work this out.
Dialogue: 0,0:00:44.34,0:00:45.29,Default,,0000,0000,0000,,When you look at this diagram,
Dialogue: 0,0:00:45.29,0:00:47.30,Default,,0000,0000,0000,,it looks like the\Nintention here on the left
Dialogue: 0,0:00:47.30,0:00:49.53,Default,,0000,0000,0000,,is this evokes the unit circle definition
Dialogue: 0,0:00:49.53,0:00:52.87,Default,,0000,0000,0000,,of trig functions because\Nthis is a unit circle
Dialogue: 0,0:00:52.87,0:00:54.98,Default,,0000,0000,0000,,right over here,
Dialogue: 0,0:00:54.98,0:00:56.35,Default,,0000,0000,0000,,and this evokes kind of\Nthe soh cah toa definition
Dialogue: 0,0:00:56.35,0:00:57.85,Default,,0000,0000,0000,,because we're just kind of in a
Dialogue: 0,0:00:57.85,0:01:00.34,Default,,0000,0000,0000,,plain, vanilla right triangle.
Dialogue: 0,0:01:00.34,0:01:02.47,Default,,0000,0000,0000,,Just to remind ourselves,
Dialogue: 0,0:01:02.47,0:01:04.26,Default,,0000,0000,0000,,let's just remind ourselves of soh cah toa
Dialogue: 0,0:01:04.26,0:01:06.17,Default,,0000,0000,0000,,because I have a feeling\Nit might be useful.
Dialogue: 0,0:01:06.17,0:01:08.60,Default,,0000,0000,0000,,Sine is opposite over hypotenuse.
Dialogue: 0,0:01:08.60,0:01:12.58,Default,,0000,0000,0000,,Cosine is adjacent over hypotenuse,
Dialogue: 0,0:01:12.58,0:01:16.60,Default,,0000,0000,0000,,and tangent is opposite over adjacent.
Dialogue: 0,0:01:16.60,0:01:18.35,Default,,0000,0000,0000,,We can refer to this and we can also
Dialogue: 0,0:01:18.35,0:01:20.97,Default,,0000,0000,0000,,remind ourselves of the\Nunit circle definition
Dialogue: 0,0:01:20.97,0:01:24.44,Default,,0000,0000,0000,,of trig functions that\Nthe cosine of an angle
Dialogue: 0,0:01:24.44,0:01:26.81,Default,,0000,0000,0000,,is the X coordinate and that the sine
Dialogue: 0,0:01:26.81,0:01:30.82,Default,,0000,0000,0000,,of where this ray\Nintersects the unit circle,
Dialogue: 0,0:01:30.82,0:01:32.41,Default,,0000,0000,0000,,and the sine of this angle is going to be
Dialogue: 0,0:01:32.41,0:01:33.58,Default,,0000,0000,0000,,the Y coordinate.
Dialogue: 0,0:01:33.58,0:01:34.85,Default,,0000,0000,0000,,What we'll see through this video
Dialogue: 0,0:01:34.85,0:01:37.50,Default,,0000,0000,0000,,is that they actually, the\Nunit circle definition,
Dialogue: 0,0:01:37.50,0:01:40.83,Default,,0000,0000,0000,,is just an extension of soh cah toa.
Dialogue: 0,0:01:40.83,0:01:44.96,Default,,0000,0000,0000,,Let's look first at X over one.
Dialogue: 0,0:01:44.96,0:01:47.89,Default,,0000,0000,0000,,We have X, X is the X coordinate.
Dialogue: 0,0:01:47.89,0:01:49.38,Default,,0000,0000,0000,,That's also the length of this side
Dialogue: 0,0:01:49.38,0:01:50.89,Default,,0000,0000,0000,,right over here,
Dialogue: 0,0:01:50.89,0:01:53.86,Default,,0000,0000,0000,,relative to this angle, theta.
Dialogue: 0,0:01:53.86,0:01:56.94,Default,,0000,0000,0000,,That is the adjacent side.
Dialogue: 0,0:01:56.94,0:01:59.15,Default,,0000,0000,0000,,So X is equal to the adjacent side.
Dialogue: 0,0:01:59.15,0:02:00.28,Default,,0000,0000,0000,,What is one?
Dialogue: 0,0:02:00.28,0:02:02.75,Default,,0000,0000,0000,,Well, this is a unit circle.
Dialogue: 0,0:02:02.75,0:02:03.87,Default,,0000,0000,0000,,One is the length of the radius
Dialogue: 0,0:02:03.87,0:02:07.19,Default,,0000,0000,0000,,which for this right triangle\Nis also the hypotenuse.
Dialogue: 0,0:02:07.19,0:02:09.40,Default,,0000,0000,0000,,If we apply the soh cah toa definition,
Dialogue: 0,0:02:09.40,0:02:12.96,Default,,0000,0000,0000,,X over one is adjacent over hypotenuse,
Dialogue: 0,0:02:12.96,0:02:15.50,Default,,0000,0000,0000,,adjacent over hypotenuse,
Dialogue: 0,0:02:15.50,0:02:18.83,Default,,0000,0000,0000,,adjacent over hypotenuse, that's cosine.
Dialogue: 0,0:02:18.83,0:02:23.89,Default,,0000,0000,0000,,That's going to be this is\Nequal to cosine of theta,
Dialogue: 0,0:02:23.89,0:02:26.27,Default,,0000,0000,0000,,but theta is the same thing as angle MKJ.
Dialogue: 0,0:02:26.27,0:02:27.59,Default,,0000,0000,0000,,They have the same measure so cosine
Dialogue: 0,0:02:27.59,0:02:31.20,Default,,0000,0000,0000,,of angle MKJ is equal to cosine of theta
Dialogue: 0,0:02:31.20,0:02:36.22,Default,,0000,0000,0000,,which is equal to X over one.
Dialogue: 0,0:02:36.22,0:02:38.34,Default,,0000,0000,0000,,Now let's move over to Y over one.
Dialogue: 0,0:02:38.34,0:02:42.15,Default,,0000,0000,0000,,Well, Y is going to be\Nthe length of this side
Dialogue: 0,0:02:42.15,0:02:43.69,Default,,0000,0000,0000,,right over here.
Dialogue: 0,0:02:43.69,0:02:47.60,Default,,0000,0000,0000,,Y is going to be, let\Nme do this in the blue.
Dialogue: 0,0:02:47.60,0:02:50.82,Default,,0000,0000,0000,,Y is going to be this length\Nrelative to angle theta.
Dialogue: 0,0:02:50.82,0:02:52.55,Default,,0000,0000,0000,,That is the opposite side.
Dialogue: 0,0:02:52.55,0:02:54.83,Default,,0000,0000,0000,,That is the opposite side.
Dialogue: 0,0:02:54.83,0:02:59.76,Default,,0000,0000,0000,,Now which trig function is\Nopposite over hypotenuse?
Dialogue: 0,0:02:59.76,0:03:01.36,Default,,0000,0000,0000,,Opposite over hypotenuse?
Dialogue: 0,0:03:01.36,0:03:03.78,Default,,0000,0000,0000,,That's sine of theta.
Dialogue: 0,0:03:03.78,0:03:05.27,Default,,0000,0000,0000,,Sine of theta.
Dialogue: 0,0:03:05.27,0:03:08.47,Default,,0000,0000,0000,,So sine of angle MKJ is the same thing
Dialogue: 0,0:03:08.47,0:03:10.34,Default,,0000,0000,0000,,as sine of theta.
Dialogue: 0,0:03:10.34,0:03:12.23,Default,,0000,0000,0000,,We see that they have the same measure,
Dialogue: 0,0:03:12.23,0:03:14.42,Default,,0000,0000,0000,,and now we see that's the same thing
Dialogue: 0,0:03:14.42,0:03:16.45,Default,,0000,0000,0000,,as Y over one.
Dialogue: 0,0:03:16.45,0:03:18.64,Default,,0000,0000,0000,,Now for both of these I used\Nthe soh cah toa definition,
Dialogue: 0,0:03:18.64,0:03:20.99,Default,,0000,0000,0000,,but we could have also used\Nthe unit circle definition.
Dialogue: 0,0:03:20.99,0:03:23.33,Default,,0000,0000,0000,,X over one, that's the same thing.
Dialogue: 0,0:03:23.33,0:03:25.53,Default,,0000,0000,0000,,That's the same thing as X,
Dialogue: 0,0:03:25.53,0:03:26.94,Default,,0000,0000,0000,,and the unit circle definition says
Dialogue: 0,0:03:26.94,0:03:30.25,Default,,0000,0000,0000,,the X coordinate of where this,
Dialogue: 0,0:03:30.25,0:03:32.45,Default,,0000,0000,0000,,I guess you could say, the\Nterminal side of this angle,
Dialogue: 0,0:03:32.45,0:03:33.70,Default,,0000,0000,0000,,this ray right over here,
Dialogue: 0,0:03:33.70,0:03:34.93,Default,,0000,0000,0000,,intersects the unit circle.
Dialogue: 0,0:03:34.93,0:03:37.67,Default,,0000,0000,0000,,That by definition, by\Nthe unit circle definition
Dialogue: 0,0:03:37.67,0:03:40.46,Default,,0000,0000,0000,,is the cosine of this angle.
Dialogue: 0,0:03:40.46,0:03:42.44,Default,,0000,0000,0000,,X is equal to the cosine of this angle,
Dialogue: 0,0:03:42.44,0:03:44.39,Default,,0000,0000,0000,,and the unit circle definition,
Dialogue: 0,0:03:44.39,0:03:48.64,Default,,0000,0000,0000,,the Y coordinate is equal\Nto the sine of this angle.
Dialogue: 0,0:03:48.64,0:03:52.34,Default,,0000,0000,0000,,We could have written\Nthis as instead of X, Y,
Dialogue: 0,0:03:52.34,0:03:54.75,Default,,0000,0000,0000,,we could have written\Nthis as cosine of theta,
Dialogue: 0,0:03:54.75,0:04:00.60,Default,,0000,0000,0000,,sine theta just like that,\Nbut let's keep going.
Dialogue: 0,0:04:00.60,0:04:02.42,Default,,0000,0000,0000,,Now we have X over Y.
Dialogue: 0,0:04:02.42,0:04:04.31,Default,,0000,0000,0000,,We have adjacent over,
Dialogue: 0,0:04:04.31,0:04:06.23,Default,,0000,0000,0000,,we have adjacent over opposite.
Dialogue: 0,0:04:06.23,0:04:10.80,Default,,0000,0000,0000,,So this is equal to\Nadjacent over opposite.
Dialogue: 0,0:04:10.80,0:04:12.85,Default,,0000,0000,0000,,Tangent is opposite over adjacent,
Dialogue: 0,0:04:12.85,0:04:14.40,Default,,0000,0000,0000,,not adjacent over opposite.
Dialogue: 0,0:04:14.40,0:04:17.11,Default,,0000,0000,0000,,This is the reciprocal of tangent.
Dialogue: 0,0:04:17.11,0:04:19.42,Default,,0000,0000,0000,,This right over here, if we had to,
Dialogue: 0,0:04:19.42,0:04:22.32,Default,,0000,0000,0000,,this is equal to one\Nover tangent of theta.
Dialogue: 0,0:04:22.32,0:04:24.62,Default,,0000,0000,0000,,We later learn about\Ncotangent and all of that
Dialogue: 0,0:04:24.62,0:04:25.55,Default,,0000,0000,0000,,which is essentially this,
Dialogue: 0,0:04:25.55,0:04:26.91,Default,,0000,0000,0000,,but it's not one of our choices.
Dialogue: 0,0:04:26.91,0:04:29.42,Default,,0000,0000,0000,,So we can rule this one out.
Dialogue: 0,0:04:29.42,0:04:31.24,Default,,0000,0000,0000,,Then we have Y over X.
Dialogue: 0,0:04:31.24,0:04:33.33,Default,,0000,0000,0000,,Well, this is looking good.
Dialogue: 0,0:04:33.33,0:04:37.20,Default,,0000,0000,0000,,This is Y is opposite.
Dialogue: 0,0:04:37.20,0:04:38.57,Default,,0000,0000,0000,,Opposite.
Dialogue: 0,0:04:38.57,0:04:41.81,Default,,0000,0000,0000,,X is adjacent relative to angle theta.
Dialogue: 0,0:04:41.81,0:04:42.74,Default,,0000,0000,0000,,Adjacent.
Dialogue: 0,0:04:42.74,0:04:45.33,Default,,0000,0000,0000,,So this is the tangent of theta.
Dialogue: 0,0:04:45.33,0:04:47.58,Default,,0000,0000,0000,,This is equal to tangent of theta.
Dialogue: 0,0:04:47.58,0:04:50.44,Default,,0000,0000,0000,,Tangent of angle MKJ is the same thing
Dialogue: 0,0:04:50.44,0:04:54.21,Default,,0000,0000,0000,,as tangent of theta which is equal,
Dialogue: 0,0:04:54.21,0:04:58.19,Default,,0000,0000,0000,,which is equal to Y over X.
Dialogue: 0,0:04:58.19,0:05:02.27,Default,,0000,0000,0000,,Now let's look at J over K, so J over K.
Dialogue: 0,0:05:02.27,0:05:03.42,Default,,0000,0000,0000,,Now we're moving over to this triangle,
Dialogue: 0,0:05:03.42,0:05:05.05,Default,,0000,0000,0000,,J over K.
Dialogue: 0,0:05:05.05,0:05:06.38,Default,,0000,0000,0000,,Relative to this angle because this
Dialogue: 0,0:05:06.38,0:05:07.53,Default,,0000,0000,0000,,is the angle that we care about,
Dialogue: 0,0:05:07.53,0:05:11.14,Default,,0000,0000,0000,,J is the length of the adjacent side,
Dialogue: 0,0:05:11.14,0:05:15.59,Default,,0000,0000,0000,,and K is the length of the opposite side,
Dialogue: 0,0:05:15.59,0:05:16.71,Default,,0000,0000,0000,,of the opposite side.
Dialogue: 0,0:05:16.71,0:05:19.17,Default,,0000,0000,0000,,This is adjacent over opposite.
Dialogue: 0,0:05:19.17,0:05:23.33,Default,,0000,0000,0000,,This is equal to adjacent over opposite.
Dialogue: 0,0:05:23.33,0:05:24.80,Default,,0000,0000,0000,,Tangent is opposite over adjacent
Dialogue: 0,0:05:24.80,0:05:25.96,Default,,0000,0000,0000,,not adjacent over opposite.
Dialogue: 0,0:05:25.96,0:05:28.100,Default,,0000,0000,0000,,So once again this is the reciprocal
Dialogue: 0,0:05:28.100,0:05:31.75,Default,,0000,0000,0000,,of the tangent function\Nnot one of the choices
Dialogue: 0,0:05:31.75,0:05:34.41,Default,,0000,0000,0000,,right over here so we\Ncan rule that one out.
Dialogue: 0,0:05:34.41,0:05:36.36,Default,,0000,0000,0000,,Now K over J.
Dialogue: 0,0:05:36.36,0:05:38.56,Default,,0000,0000,0000,,Well, now this is opposite over adjacent.
Dialogue: 0,0:05:38.56,0:05:40.24,Default,,0000,0000,0000,,Opposite over adjacent.
Dialogue: 0,0:05:40.24,0:05:42.93,Default,,0000,0000,0000,,That is equal to tangent of theta.
Dialogue: 0,0:05:42.93,0:05:45.42,Default,,0000,0000,0000,,This is equal to tangent of theta,
Dialogue: 0,0:05:45.42,0:05:47.58,Default,,0000,0000,0000,,or tangent of angle MKJ.
Dialogue: 0,0:05:47.58,0:05:51.90,Default,,0000,0000,0000,,This is equal to K over J.
Dialogue: 0,0:05:51.90,0:05:55.60,Default,,0000,0000,0000,,Now we have M over J, M over J.
Dialogue: 0,0:05:55.60,0:05:58.81,Default,,0000,0000,0000,,Hypotenuse over adjacent side.
Dialogue: 0,0:05:58.81,0:06:02.75,Default,,0000,0000,0000,,This of course is equal to the hypotenuse.
Dialogue: 0,0:06:02.75,0:06:04.95,Default,,0000,0000,0000,,Hypotenuse over adjacent.
Dialogue: 0,0:06:04.95,0:06:06.48,Default,,0000,0000,0000,,Well, if it was adjacent over hypotenuse,
Dialogue: 0,0:06:06.48,0:06:07.42,Default,,0000,0000,0000,,we'd be dealing with cosine,
Dialogue: 0,0:06:07.42,0:06:09.19,Default,,0000,0000,0000,,but this is the reciprocal of that.
Dialogue: 0,0:06:09.19,0:06:12.80,Default,,0000,0000,0000,,This is actually one\Nover the cosine of theta
Dialogue: 0,0:06:12.80,0:06:14.29,Default,,0000,0000,0000,,not one of our choices,
Dialogue: 0,0:06:14.29,0:06:15.79,Default,,0000,0000,0000,,not one of our choices here so I'll just
Dialogue: 0,0:06:15.79,0:06:17.72,Default,,0000,0000,0000,,rule that one out over there.
Dialogue: 0,0:06:17.72,0:06:20.39,Default,,0000,0000,0000,,Then we have it's reciprocal, J over M.
Dialogue: 0,0:06:20.39,0:06:22.53,Default,,0000,0000,0000,,That's adjacent over hypotenuse.
Dialogue: 0,0:06:22.53,0:06:25.60,Default,,0000,0000,0000,,Adjacent over hypotenuse is cosine.
Dialogue: 0,0:06:25.60,0:06:27.60,Default,,0000,0000,0000,,This is equal to cosine of theta,
Dialogue: 0,0:06:27.60,0:06:31.15,Default,,0000,0000,0000,,or cosine of angle MKJ so\Nwe could write it down.
Dialogue: 0,0:06:31.15,0:06:35.20,Default,,0000,0000,0000,,This is equivalent to J over M.
Dialogue: 0,0:06:35.20,0:06:37.44,Default,,0000,0000,0000,,Then one last one, K over M.
Dialogue: 0,0:06:37.44,0:06:39.48,Default,,0000,0000,0000,,Well, that's opposite over hypotenuse,
Dialogue: 0,0:06:39.48,0:06:40.94,Default,,0000,0000,0000,,opposite over hypotenuse.
Dialogue: 0,0:06:40.94,0:06:43.54,Default,,0000,0000,0000,,That's going to be sine of theta.
Dialogue: 0,0:06:43.54,0:06:46.52,Default,,0000,0000,0000,,This right over here is\Nequal to sine of theta
Dialogue: 0,0:06:46.52,0:06:48.64,Default,,0000,0000,0000,,which is the same thing\Nas sine of angle MKJ
Dialogue: 0,0:06:48.64,0:06:50.53,Default,,0000,0000,0000,,which is the same thing as\Nall of these expressions.
Dialogue: 0,0:06:50.53,0:06:54.12,Default,,0000,0000,0000,,This is equal to K over M,
Dialogue: 0,0:06:54.12,0:06:55.88,Default,,0000,0000,0000,,and we are done.