WEBVTT 00:00:01.143 --> 00:00:02.735 Voiceover:Artemis seeks knowledge of 00:00:02.735 --> 00:00:04.602 the width of Orion's belt, 00:00:04.602 --> 00:00:08.101 which is a pattern of stars in the Orion constellation. 00:00:08.101 --> 00:00:10.741 She has previously discovered the distances 00:00:10.741 --> 00:00:16.804 from her house to Alnitak, 736 lights years, 00:00:16.804 --> 00:00:20.756 and to Mintaka, 915 light years, 00:00:20.756 --> 00:00:23.758 which are the endpoints of Orion's belt. 00:00:23.758 --> 00:00:26.002 She knows the angle between these 00:00:26.002 --> 00:00:29.142 stars in the sky is three degrees. 00:00:29.142 --> 00:00:31.806 What is the width of Orion's belt? 00:00:31.806 --> 00:00:32.872 That is, what is the distance 00:00:32.872 --> 00:00:36.268 between Alnitak and Mintaka? 00:00:36.268 --> 00:00:39.209 And they want us to the answer in light years. 00:00:39.209 --> 00:00:40.872 So let's draw a little diagram 00:00:40.872 --> 00:00:42.607 to make sure we understand what's going on. 00:00:42.607 --> 00:00:43.670 Actually, even before we do that, 00:00:43.670 --> 00:00:44.563 I encourage you to pause 00:00:44.563 --> 00:00:46.735 this and try this on your own. 00:00:46.735 --> 00:00:48.675 Now let's make a diagram. 00:00:48.675 --> 00:00:50.798 Alright, so let's say that this is Artemis' 00:00:50.798 --> 00:00:52.141 house right over here. 00:00:52.141 --> 00:00:53.756 This is Artemis' house. 00:00:53.756 --> 00:00:57.174 I'll say that's A for Artemis' house. 00:00:57.174 --> 00:00:58.666 And then... 00:00:58.666 --> 00:01:00.004 Alright, let me say H... 00:01:00.004 --> 00:01:01.605 Let me say this is home. 00:01:01.605 --> 00:01:03.273 This is home right over here. 00:01:03.273 --> 00:01:04.748 And we have these 2 stars. 00:01:04.748 --> 00:01:07.467 So she's looking out into the night sky 00:01:07.467 --> 00:01:09.169 and she sees these stars, 00:01:09.169 --> 00:01:14.605 Alnitak, which is 736 light years away, 00:01:14.605 --> 00:01:17.337 and obviously I'm not going to draw this to scale. 00:01:17.337 --> 00:01:21.750 So this is Alnitak. 00:01:21.750 --> 00:01:25.521 And Mintaka. 00:01:25.521 --> 00:01:28.606 So let's say this is Mintaka right over here. 00:01:28.606 --> 00:01:31.001 Mintaka. 00:01:31.001 --> 00:01:32.606 And we know a few things. 00:01:32.606 --> 00:01:35.273 We know that this distance between her home 00:01:35.273 --> 00:01:40.173 and Alnitak is 736 light years. 00:01:40.173 --> 00:01:42.935 So this distance right over here. 00:01:42.935 --> 00:01:44.338 So that right over there. 00:01:44.338 --> 00:01:45.837 Everything we'll do is in light years. 00:01:45.837 --> 00:01:47.707 That's 736. 00:01:47.707 --> 00:01:48.605 And the distance between 00:01:48.605 --> 00:01:54.674 her house and Mintaka is 915 light years. 00:01:54.674 --> 00:01:57.163 So it would take light 915 years 00:01:57.163 --> 00:01:58.879 to get from her house to Mintaka, 00:01:58.879 --> 00:02:01.248 or from Mintaka to her house. 00:02:01.248 --> 00:02:04.271 So this is 915 light years. 00:02:04.271 --> 00:02:05.379 And what we wanna do is figure out 00:02:05.379 --> 00:02:07.402 the width of Orion's belt, 00:02:07.402 --> 00:02:11.136 which is the distance between Alnitak and Mintaka. 00:02:11.136 --> 00:02:15.880 So we need to figure out this distance 00:02:15.880 --> 00:02:17.335 right over here. 00:02:17.335 --> 00:02:21.506 And the one thing that they did give us 00:02:21.506 --> 00:02:23.269 is this angle. 00:02:23.269 --> 00:02:26.270 They did give us that angle right over there. 00:02:26.270 --> 00:02:28.136 They said that the angle between 00:02:28.136 --> 00:02:30.216 these stars in the sky is three degrees. 00:02:30.216 --> 00:02:33.552 So this is three degrees right over there. 00:02:33.552 --> 00:02:36.003 So how can we figure out the distance 00:02:36.003 --> 00:02:38.406 between Alnitak and Mintaka? 00:02:38.406 --> 00:02:40.868 Let's just say that this is equal to X. 00:02:40.868 --> 00:02:42.074 This is equal to X. 00:02:42.074 --> 00:02:43.404 How do we do that? 00:02:43.404 --> 00:02:45.697 Well if we have two sides 00:02:45.697 --> 00:02:47.990 and an angle between them, 00:02:47.990 --> 00:02:50.285 we could use the law of cosines 00:02:50.285 --> 00:02:55.368 to figure out the third side. 00:02:55.368 --> 00:02:56.736 So the law of cosines, 00:02:56.736 --> 00:02:58.534 so let's just apply it. 00:02:58.534 --> 00:03:02.871 So the law of cosines tells us 00:03:02.871 --> 00:03:05.928 that X squared is going to be equal 00:03:05.928 --> 00:03:09.176 to the sum of the squares of the other two sides. 00:03:09.176 --> 00:03:14.433 So it's going to be equal to 736 squared, 00:03:14.433 --> 00:03:28.534 plus 915 squared, minus two times 736, 00:03:28.534 --> 00:03:37.135 times 915, times the cosine of this angle. 00:03:37.135 --> 00:03:41.631 Times the cosine of three degrees. 00:03:41.631 --> 00:03:43.473 So once again, 00:03:43.473 --> 00:03:44.541 we're trying to find the length of 00:03:44.541 --> 00:03:46.501 the side opposite the three degrees. 00:03:46.501 --> 00:03:48.008 We know the other two sides, 00:03:48.008 --> 00:03:50.084 so the law of cosines, it essentially... 00:03:51.807 --> 00:03:54.061 Sorry, I just had to cough off camera 00:03:54.061 --> 00:03:56.211 because I had some peanuts and my throat was dry. 00:03:56.211 --> 00:03:56.997 Where was I? 00:03:56.997 --> 00:03:58.324 Oh, I was saying, 00:03:58.324 --> 00:04:00.543 if we know the angle and we know the two sides 00:04:00.543 --> 00:04:01.799 on either side of the angle, 00:04:01.799 --> 00:04:03.294 we can figure out the length of the side opposite 00:04:03.294 --> 00:04:04.853 by the law of cosines. 00:04:04.853 --> 00:04:06.624 Where it essentially starts off not too 00:04:06.624 --> 00:04:08.215 different than the Pythagorean theorem, 00:04:08.215 --> 00:04:09.330 but then we give an adjustment 00:04:09.330 --> 00:04:12.210 because this is not an actual right triangle. 00:04:12.210 --> 00:04:13.264 And the adjustment... 00:04:13.264 --> 00:04:16.262 So we have the 736 squared, plus 915 squared, 00:04:16.262 --> 00:04:19.426 minus two times the product of these sides, 00:04:19.426 --> 00:04:21.674 times the cosine of this angle. 00:04:21.674 --> 00:04:23.669 Or another way we could say, think about it is, 00:04:23.669 --> 00:04:28.722 X, let me write that, 00:04:28.722 --> 00:04:31.591 X is to equal to the square root of all 00:04:31.591 --> 00:04:33.000 of this stuff. 00:04:33.000 --> 00:04:36.174 So, I can just copy and paste that. 00:04:37.482 --> 00:04:39.115 Copy and paste. 00:04:40.126 --> 00:04:44.487 X is going to be equal to the square root of that. 00:04:44.886 --> 00:04:48.328 And so let's get our calculator to calculate it. 00:04:48.328 --> 00:04:51.054 And let me verify that I'm in degree mode. 00:04:51.054 --> 00:04:53.727 Yes, I am indeed in degree mode. 00:04:53.727 --> 00:04:55.724 And so let's exit that. 00:04:55.724 --> 00:04:58.975 And so I wanna calculate the square root of 00:04:58.975 --> 00:05:06.641 736 squared, plus 915 squared, 00:05:06.641 --> 00:05:15.993 minus two times 736, times 915, 00:05:15.993 --> 00:05:19.657 times cosine of three degrees. 00:05:19.657 --> 00:05:22.474 And we deserve a drum roll now. 00:05:22.474 --> 00:05:24.552 X is 100, if we round... 00:05:24.552 --> 00:05:25.785 Let's see, what did they want us to do? 00:05:25.785 --> 00:05:27.552 Round your answer to the nearest light years. 00:05:27.552 --> 00:05:28.460 So to the nearest light year 00:05:28.460 --> 00:05:31.932 is going to be 184 light years. 00:05:31.932 --> 00:05:40.590 So X is approximately equal to 184 light years. 00:05:40.590 --> 00:05:43.559 So it would take light 184 years 00:05:43.559 --> 00:05:47.970 to get from Mintaka to Alnitak. 00:05:47.970 --> 00:05:49.189 And so hopefully this actually shows you 00:05:49.189 --> 00:05:51.927 if you are going to do any astronomy, 00:05:51.927 --> 00:05:54.057 the law of cosines, law of sines, 00:05:54.057 --> 00:05:55.992 in fact all of trigonometry, 00:05:55.992 --> 00:05:59.992 becomes quite, quite handy.