0:00:01.930,0:00:02.288 Voiceover:On the right-hand side we have 0:00:02.288,0:00:04.322 a bunch of expressions[br]that are just ratios 0:00:04.322,0:00:07.514 of different information[br]given in these two diagrams. 0:00:07.514,0:00:09.832 Then over here on the left we have 0:00:09.832,0:00:11.857 the sine taken of angle MKJ, 0:00:11.857,0:00:15.196 cosine of angle MKJ, and[br]tangent of angle MKJ. 0:00:15.196,0:00:19.407 Angle MKJ is this angle right over here 0:00:19.407,0:00:22.505 same thing as theta, so these two angles. 0:00:22.505,0:00:25.189 These two angles have the same measure. 0:00:25.189,0:00:26.767 We see that right over there. 0:00:26.767,0:00:28.178 What we want to do is figure out 0:00:28.178,0:00:30.551 which of these expressions are equivalent 0:00:30.551,0:00:34.440 to which of these[br]expressions right over here. 0:00:34.440,0:00:36.706 I encourage you to pause the video 0:00:36.706,0:00:40.312 and try to work this through on your own. 0:00:40.312,0:00:42.897 Assuming you've had a go at it, 0:00:42.897,0:00:44.338 let's try to work this out. 0:00:44.338,0:00:45.287 When you look at this diagram, 0:00:45.287,0:00:47.300 it looks like the[br]intention here on the left 0:00:47.300,0:00:49.531 is this evokes the unit circle definition 0:00:49.531,0:00:52.868 of trig functions because[br]this is a unit circle 0:00:52.868,0:00:54.980 right over here, 0:00:54.980,0:00:56.350 and this evokes kind of[br]the soh cah toa definition 0:00:56.350,0:00:57.849 because we're just kind of in a 0:00:57.849,0:01:00.337 plain, vanilla right triangle. 0:01:00.337,0:01:02.470 Just to remind ourselves, 0:01:02.470,0:01:04.260 let's just remind ourselves of soh cah toa 0:01:04.260,0:01:06.172 because I have a feeling[br]it might be useful. 0:01:06.172,0:01:08.603 Sine is opposite over hypotenuse. 0:01:08.603,0:01:12.577 Cosine is adjacent over hypotenuse, 0:01:12.577,0:01:16.603 and tangent is opposite over adjacent. 0:01:16.603,0:01:18.350 We can refer to this and we can also 0:01:18.350,0:01:20.971 remind ourselves of the[br]unit circle definition 0:01:20.971,0:01:24.435 of trig functions that[br]the cosine of an angle 0:01:24.435,0:01:26.809 is the X coordinate and that the sine 0:01:26.809,0:01:30.816 of where this ray[br]intersects the unit circle, 0:01:30.816,0:01:32.407 and the sine of this angle is going to be 0:01:32.407,0:01:33.578 the Y coordinate. 0:01:33.578,0:01:34.849 What we'll see through this video 0:01:34.849,0:01:37.497 is that they actually, the[br]unit circle definition, 0:01:37.497,0:01:40.829 is just an extension of soh cah toa. 0:01:40.829,0:01:44.965 Let's look first at X over one. 0:01:44.965,0:01:47.892 We have X, X is the X coordinate. 0:01:47.892,0:01:49.377 That's also the length of this side 0:01:49.377,0:01:50.891 right over here, 0:01:50.891,0:01:53.855 relative to this angle, theta. 0:01:53.855,0:01:56.940 That is the adjacent side. 0:01:56.940,0:01:59.150 So X is equal to the adjacent side. 0:01:59.150,0:02:00.285 What is one? 0:02:00.285,0:02:02.750 Well, this is a unit circle. 0:02:02.750,0:02:03.867 One is the length of the radius 0:02:03.867,0:02:07.190 which for this right triangle[br]is also the hypotenuse. 0:02:07.190,0:02:09.398 If we apply the soh cah toa definition, 0:02:09.398,0:02:12.963 X over one is adjacent over hypotenuse, 0:02:12.963,0:02:15.495 adjacent over hypotenuse, 0:02:15.495,0:02:18.826 adjacent over hypotenuse, that's cosine. 0:02:18.826,0:02:23.888 That's going to be this is[br]equal to cosine of theta, 0:02:23.888,0:02:26.272 but theta is the same thing as angle MKJ. 0:02:26.272,0:02:27.594 They have the same measure so cosine 0:02:27.594,0:02:31.198 of angle MKJ is equal to cosine of theta 0:02:31.198,0:02:36.219 which is equal to X over one. 0:02:36.219,0:02:38.339 Now let's move over to Y over one. 0:02:38.339,0:02:42.150 Well, Y is going to be[br]the length of this side 0:02:42.150,0:02:43.686 right over here. 0:02:43.686,0:02:47.599 Y is going to be, let[br]me do this in the blue. 0:02:47.599,0:02:50.825 Y is going to be this length[br]relative to angle theta. 0:02:50.825,0:02:52.553 That is the opposite side. 0:02:52.553,0:02:54.831 That is the opposite side. 0:02:54.831,0:02:59.764 Now which trig function is[br]opposite over hypotenuse? 0:02:59.764,0:03:01.361 Opposite over hypotenuse? 0:03:01.361,0:03:03.780 That's sine of theta. 0:03:03.780,0:03:05.274 Sine of theta. 0:03:05.274,0:03:08.470 So sine of angle MKJ is the same thing 0:03:08.470,0:03:10.341 as sine of theta. 0:03:10.341,0:03:12.230 We see that they have the same measure, 0:03:12.230,0:03:14.425 and now we see that's the same thing 0:03:14.425,0:03:16.453 as Y over one. 0:03:16.453,0:03:18.643 Now for both of these I used[br]the soh cah toa definition, 0:03:18.643,0:03:20.994 but we could have also used[br]the unit circle definition. 0:03:20.994,0:03:23.329 X over one, that's the same thing. 0:03:23.329,0:03:25.530 That's the same thing as X, 0:03:25.530,0:03:26.942 and the unit circle definition says 0:03:26.942,0:03:30.249 the X coordinate of where this, 0:03:30.249,0:03:32.446 I guess you could say, the[br]terminal side of this angle, 0:03:32.446,0:03:33.704 this ray right over here, 0:03:33.704,0:03:34.926 intersects the unit circle. 0:03:34.926,0:03:37.674 That by definition, by[br]the unit circle definition 0:03:37.674,0:03:40.455 is the cosine of this angle. 0:03:40.455,0:03:42.444 X is equal to the cosine of this angle, 0:03:42.444,0:03:44.390 and the unit circle definition, 0:03:44.390,0:03:48.640 the Y coordinate is equal[br]to the sine of this angle. 0:03:48.640,0:03:52.339 We could have written[br]this as instead of X, Y, 0:03:52.339,0:03:54.746 we could have written[br]this as cosine of theta, 0:03:54.746,0:04:00.600 sine theta just like that,[br]but let's keep going. 0:04:00.600,0:04:02.419 Now we have X over Y. 0:04:02.419,0:04:04.312 We have adjacent over, 0:04:04.312,0:04:06.229 we have adjacent over opposite. 0:04:06.229,0:04:10.797 So this is equal to[br]adjacent over opposite. 0:04:10.797,0:04:12.853 Tangent is opposite over adjacent, 0:04:12.853,0:04:14.404 not adjacent over opposite. 0:04:14.404,0:04:17.108 This is the reciprocal of tangent. 0:04:17.108,0:04:19.425 This right over here, if we had to, 0:04:19.425,0:04:22.322 this is equal to one[br]over tangent of theta. 0:04:22.322,0:04:24.622 We later learn about[br]cotangent and all of that 0:04:24.622,0:04:25.552 which is essentially this, 0:04:25.552,0:04:26.906 but it's not one of our choices. 0:04:26.906,0:04:29.425 So we can rule this one out. 0:04:29.425,0:04:31.240 Then we have Y over X. 0:04:31.240,0:04:33.333 Well, this is looking good. 0:04:33.333,0:04:37.195 This is Y is opposite. 0:04:37.195,0:04:38.574 Opposite. 0:04:38.574,0:04:41.812 X is adjacent relative to angle theta. 0:04:41.812,0:04:42.745 Adjacent. 0:04:42.745,0:04:45.328 So this is the tangent of theta. 0:04:45.328,0:04:47.585 This is equal to tangent of theta. 0:04:47.585,0:04:50.440 Tangent of angle MKJ is the same thing 0:04:50.440,0:04:54.212 as tangent of theta which is equal, 0:04:54.212,0:04:58.187 which is equal to Y over X. 0:04:58.187,0:05:02.267 Now let's look at J over K, so J over K. 0:05:02.267,0:05:03.419 Now we're moving over to this triangle, 0:05:03.419,0:05:05.051 J over K. 0:05:05.051,0:05:06.381 Relative to this angle because this 0:05:06.381,0:05:07.533 is the angle that we care about, 0:05:07.533,0:05:11.136 J is the length of the adjacent side, 0:05:11.136,0:05:15.593 and K is the length of the opposite side, 0:05:15.593,0:05:16.706 of the opposite side. 0:05:16.706,0:05:19.172 This is adjacent over opposite. 0:05:19.172,0:05:23.330 This is equal to adjacent over opposite. 0:05:23.330,0:05:24.804 Tangent is opposite over adjacent 0:05:24.804,0:05:25.960 not adjacent over opposite. 0:05:25.960,0:05:28.999 So once again this is the reciprocal 0:05:28.999,0:05:31.750 of the tangent function[br]not one of the choices 0:05:31.750,0:05:34.407 right over here so we[br]can rule that one out. 0:05:34.407,0:05:36.357 Now K over J. 0:05:36.357,0:05:38.565 Well, now this is opposite over adjacent. 0:05:38.565,0:05:40.241 Opposite over adjacent. 0:05:40.241,0:05:42.927 That is equal to tangent of theta. 0:05:42.927,0:05:45.418 This is equal to tangent of theta, 0:05:45.418,0:05:47.580 or tangent of angle MKJ. 0:05:47.580,0:05:51.900 This is equal to K over J. 0:05:51.900,0:05:55.605 Now we have M over J, M over J. 0:05:55.605,0:05:58.809 Hypotenuse over adjacent side. 0:05:58.809,0:06:02.748 This of course is equal to the hypotenuse. 0:06:02.748,0:06:04.953 Hypotenuse over adjacent. 0:06:04.953,0:06:06.482 Well, if it was adjacent over hypotenuse, 0:06:06.482,0:06:07.415 we'd be dealing with cosine, 0:06:07.415,0:06:09.187 but this is the reciprocal of that. 0:06:09.187,0:06:12.795 This is actually one[br]over the cosine of theta 0:06:12.795,0:06:14.288 not one of our choices, 0:06:14.288,0:06:15.786 not one of our choices here so I'll just 0:06:15.786,0:06:17.723 rule that one out over there. 0:06:17.723,0:06:20.393 Then we have it's reciprocal, J over M. 0:06:20.393,0:06:22.534 That's adjacent over hypotenuse. 0:06:22.534,0:06:25.599 Adjacent over hypotenuse is cosine. 0:06:25.599,0:06:27.604 This is equal to cosine of theta, 0:06:27.604,0:06:31.152 or cosine of angle MKJ so[br]we could write it down. 0:06:31.152,0:06:35.196 This is equivalent to J over M. 0:06:35.196,0:06:37.438 Then one last one, K over M. 0:06:37.438,0:06:39.479 Well, that's opposite over hypotenuse, 0:06:39.479,0:06:40.940 opposite over hypotenuse. 0:06:40.940,0:06:43.545 That's going to be sine of theta. 0:06:43.545,0:06:46.517 This right over here is[br]equal to sine of theta 0:06:46.517,0:06:48.641 which is the same thing[br]as sine of angle MKJ 0:06:48.641,0:06:50.528 which is the same thing as[br]all of these expressions. 0:06:50.528,0:06:54.120 This is equal to K over M, 0:06:54.120,0:06:55.881 and we are done.