WEBVTT 00:00:07.350 --> 00:00:14.856 Let's look at how we can solve an oblique triangle. An oblique triangle is a triangle that does not have 00:00:14.858 --> 00:00:20.856 a right angle, so we don't have the Pythagorean theorem to look at, and we don't have the sine is opposite 00:00:20.858 --> 00:00:26.378 over the hypotenuse and that sort of thing. So we have other tools to use, and one of the tools we can use is the 00:00:26.380 --> 00:00:35.637 law of sines. In any triangle you have six parts, and you have these ratios that will always be true. The 00:00:35.639 --> 00:00:44.644 sine of alpha is to a as the sine of beta is to b as the sine of gamma is to c, or you can use the reciprocal. 00:00:44.646 --> 00:00:50.648 Now, one thing that students tend to have trouble with is when do they use the law of sines. So I'm going 00:00:50.650 --> 00:00:59.911 to show you a technique that you can use. If you set up a little chart with your sides and your angles listed and 00:00:59.913 --> 00:01:05.664 then you check off or fill in the material that you're given, the values that you're given, then you'll know. 00:01:05.666 --> 00:01:13.675 Anytime you have one side and the angle opposite that side you have a ratio. You can set up one of these 00:01:13.677 --> 00:01:20.933 ratios, and then you can find the other missing parts. Anytime you have a side and the opposite angle you 00:01:20.935 --> 00:01:28.174 can use the law of sines. There are actually three cases. We have the angle-side-angle, side-angle-angle, and 00:01:28.176 --> 00:01:35.000 side-side-angle. These are the cases that you use the law of sines to solve the triangle. 00:01:37.467 --> 00:01:43.422 Let's solve a triangle using the law of sines. First thing you always want to do is you want to organize your 00:01:43.4244 --> 00:01:50.925 data. Set up a little chart here. You're going to find that you'll be able to solve these much more efficiently 00:01:50.927 --> 00:01:56.218 because you'll immediately have everything in order. You can see what you have, what you can find 00:01:56.220 --> 00:02:03.224 quickly, and we'll go from there. So what we have here is we know that c is equal to 65. We're given that 00:02:03.226 --> 00:02:12.472 alpha is equal to 46 degrees and beta is equal to 20 degrees. Now, you might say, well, I don't have a 00:02:12.474 --> 00:02:17.974 ratio here, so how am I going to solve using the law of sines because you have to have a ratio. But we have 00:02:17.976 --> 00:02:23.745 two angles, and anytime you have two angles in a triangle you find the third angle very quickly because 00:02:23.747 --> 00:02:32.994 the sum of the angles is 180 degrees. So we're going to know that gamma is going to be 180 degrees minus 00:02:32.996 --> 00:02:43.762 46 degrees minus 20 degrees. So we have an exact value for gamma. It's 114 degrees. So that's one of our 00:02:43.764 --> 00:02:49.763 missing parts. So now we have that ratio that we said we needed to use the law of sines. So we're going 00:02:49.765 --> 00:03:03.764 to find sides a and b by setting up the ratio a is to the sine of 46 degrees. Now, remember we use what we're 00:03:03.766 --> 00:03:22.515 given, so we have that 65 is to the sine of 114 degrees. So a is equal to 65 times the sine of 46 degrees 00:03:22.517 --> 00:03:30.064 divided by the sine of 114 degrees. We'll calculate that in a minute, and we're going to do the same thing 00:03:30.066 --> 00:03:48.591 for side b. So we have this. Now we know that b is to the sine of 20 degrees as 65 is to the sine of 114 degrees, 00:03:48.591 --> 00:04:03.845 so we can solve for b. [Side] b is 65 times the sine of 20 degrees divided by the sine of 114 degrees. Now we can 00:04:03.847 --> 00:04:11.095 find a decimal approximation for these two, and we solve our triangle. Alright, make sure again that you 00:04:11.097 --> 00:04:15.111 have it in degree mode because if you've been working in radian mode and degree mode you might want 00:04:15.113 --> 00:04:23.347 to make sure that you double check that because you could get seriously wrong answers. Divide by the sine 00:04:23.616 --> 00:04:42.365 of 114 and we get that a is approximately 51.18, and now we'll find b. So multiply 65 times the sine of 20 00:04:42.367 --> 00:04:51.613 divided by the sine of 114, and oops, I got the wrong answer there. I can see that. I got sine sine in there, 00:04:51.615 --> 00:04:58.148 so we'll redo it. So we have 65. Part of the thing is in any kind of math you're going to get the wrong 00:04:58.148 --> 00:05:03.902 answer. The trick is to know when you've got the wrong answer and check your answers again. So it's not 00:05:03.904 --> 00:05:10.161 that we don't make mistakes. It's just knowing when you do. Alright, so this one works out better. This is 00:05:10.163 --> 00:05:19.660 approximately 24.34. Now I want to point out something in a triangle. I want you to notice this because 00:05:19.662 --> 00:05:24.662 this can help you in other triangles when you're solving problems make sure you've got your data 00:05:24.664 --> 00:05:30.659 organized. You'll notice that the largest side is always going to be opposite the largest angle. The 00:05:30.661 --> 00:05:35.184 smallest side is opposite the smallest angle. So you can kind of give a visual check there and make sure 00:05:35.186 --> 00:05:39.939 that you've written it down. You might find the right answer and then write it in the wrong place. So 00:05:39.941 --> 00:05:44.197 if you kind of just do a visual check of that, you'll see that you've got everything as it should be. 00:05:44.199 --> 00:05:48.650 So now we have solved that triangle. 00:05:50.445 --> 00:05:55.695 Now we're going to look at something called the ambiguous case within the law of sines, and this is when 00:05:55.697 --> 00:06:02.691 you have side-side-angle. You have an angle, the side opposite that angle, and one other side. If you set it 00:06:02.693 --> 00:06:10.006 up in your little chart, if you look at this, and you have the three sides and the three angles. If you're given 00:06:10.008 --> 00:06:18.263 one side and the angle opposite that and either of the other two sides, it doesn't matter, and this can be in 00:06:18.265 --> 00:06:23.765 any order, then that's what we call the ambiguous case. Now we call it that because you might go back and 00:06:23.767 --> 00:06:32.263 remember about inverse sines. We know that on your calculator if you enter sine inverse of 1/2 it's going 00:06:32.265 --> 00:06:40.036 to give you an answer of 30 degrees, but you have to remember that the sine of 150 degrees is also equal 00:06:40.038 --> 00:06:48.785 to 1/2. Your calculator's not going to tell you that, but 30 degrees and 150 degrees could both be angles in 00:06:48.785 --> 00:06:54.286 a triangle. They're both less than 180 degrees. So this creates what we call the ambiguous case. Now the 00:06:54.288 --> 00:07:00.784 ambiguous case has three outcomes possible. You can have zero triangles, you can have one triangle, or 00:07:00.786 --> 00:07:07.035 you can have two triangles, but I'm going to show you how you can find that out very quickly. We're going 00:07:07.037 --> 00:07:14.286 to continue with this technique, and now we'll solve an example using the law of sines, ambiguous case. 00:07:16.280 --> 00:07:21.422 Alright, as we always do, we're going to take the triangle and the information we're given, and we're 00:07:21.424 --> 00:07:30.947 going to put it in our chart. So we're told that a is equal to 3, b is equal to 2. We don't have a value for c. 00:07:30.949 --> 00:07:41.174 We're given that alpha is equal to 40 degrees, and then we have to find beta and gamma. Now, this is the 00:07:41.176 --> 00:07:47.450 ambiguous case. Remember, how do you know it's the ambiguous case? You're given one side and the 00:07:47.452 --> 00:07:54.201 angle opposite that side and one of the other two sides. So now, we look at this and we say, "Alright, 00:07:54.203 --> 00:08:00.201 what's the first thing I can find?" Well obviously I don't have any way of finding c or gamma, but I can use 00:08:00.203 --> 00:08:06.701 the law of sines to find beta. So I'm going to set up my ratio, and we're going to find beta. I know that the 00:08:06.703 --> 00:08:20.703 sine of beta is to 2, which is side b, as the sine of 40 degrees is to 3. Alright, now I'm going to solve for beta. 00:08:20.705 --> 00:08:29.994 This will take a little algebra here. We know that the sine of beta is 2 times the sine of 40 degrees divided 00:08:29.996 --> 00:08:43.257 by 3, so beta is sine inverse of 2 times the sine of 40 degrees divided by 3. So we're going to find that value, 00:08:43.259 --> 00:08:48.516 and be careful when you put this in your calculator. Make sure you put parentheses around this whole 00:08:48.518 --> 00:08:55.515 expression because otherwise you would definitely get a wrong answer. So we have sine inverse of 2 00:08:55.517 --> 00:09:06.014 times the sine of 40 divided by 3, and we get 25 point. Well I guess we'll just round it off to one decimal, 00:09:06.016 --> 00:09:15.787 so that would be 25.4 degrees, so beta is approximately 25.4 degrees. Now, I know I have at least one 00:09:15.789 --> 00:09:26.287 triangle because I found beta. So the question is how do I know if there's one triangle, just this one, or if 00:09:26.289 --> 00:09:36.310 there are two triangles. So here's what you do. Remember that the sine of 25.4 degrees is the same as the 00:09:36.312 --> 00:09:52.062 sine of the supplement to 25.4 degrees. If I find the supplement of 25.4 degrees, so we have 180 minus 25.4, 00:09:52.064 --> 00:10:07.593 that's 154.6 degrees. So the sine of 154.6 degrees is the same number in here. So the question is, okay, can 00:10:07.595 --> 00:10:16.848 that work in this triangle? Well, what would happen if I substituted 154.6 degrees in for this? Let me just 00:10:16.850 --> 00:10:22.610 erase this here, and let's put it here, and let's see what would happen. So I'm saying, ok, I found the other 00:10:22.612 --> 00:10:32.350 angle and I'm good to go. But now I look and I see, oops, I've got 154.6 degrees plus 40 degrees is greater 00:10:32.352 --> 00:10:38.372 than 180 degrees, and the sum of all of them can only be 180 degrees. So we know that that is not going to 00:10:38.374 --> 00:10:48.133 be the case and that the only value for beta in this particular set of data that we're given is 25.4 degrees, 00:10:48.135 --> 00:10:53.631 and so now we know we only have one triangle and we're going to find the other two missing parts. 00:10:53.633 --> 00:10:59.884 Well we can't find c. Remember a squared plus b squared is not equal to c squared, so we're going to find 00:10:59.886 --> 00:11:07.402 gamma using the fact that the sum of the angles of a triangle is 180. So gamma, actually we'll put 00:11:07.404 --> 00:11:20.153 approximately because we're approximating this. So gamma is approximately 180 minus 40 minus 25.4, 00:11:20.155 --> 00:11:36.901 so that's 114.6 degrees. Now we can find the third side, c. Now remember, don't use beta to find c because 00:11:36.903 --> 00:11:42.426 you've already approximated here. You've already approximated here. Use what you're given, more 00:11:42.428 --> 00:11:49.653 exact values. You'll get a better approximation. You're less likely to make a roundoff error. So we know 00:11:49.655 --> 00:12:07.153 that c is to the sine of 114.6 degrees as 3 is to the sine of 40 degrees, so we can solve for c now. So c is 00:12:07.155 --> 00:12:24.479 3 times the sine of 114.6 degrees divided by the sine of 40 degrees. So we'll calculate that, and we get 00:12:24.481 --> 00:12:33.229 4.24. Now let's go back and double check. Remember we talked about the fact that the largest angle is 00:12:33.231 --> 00:12:41.981 opposite the largest side, and we see that this is the largest angle, the largest side, and that the ratios hold. 00:12:41.983 --> 00:12:48.754 So now we have found our triangle. We've discovered that in this particular set of measurements 00:12:48.756 --> 00:12:53.550 there was only one triangle, so we have solved this oblique triangle. 00:12:56.252 --> 00:13:02.250 Let's take a look at another example of the ambiguous case. How do I know it's the ambiguous case? 00:13:02.252 --> 00:13:09.335 Because I have one side, a given angle opposite that side, and one other side. So the first thing that we're going to 00:13:09.337 --> 00:13:14.346 do is we're going to try to find alpha because that's all really that I can do. So I'm going to set up my 00:13:14.348 --> 00:13:27.843 trig ratios for the law of sines. The sine of alpha is to 2 as the sine of 60 degrees is to 1, so the sine of alpha 00:13:27.845 --> 00:13:38.096 is 2 times the sine of 60 degrees. Now you might already see what's happening here, but just in case 00:13:38.098 --> 00:13:48.347 you don't, or it's an angle you're maybe not familiar with, alpha is sine inverse of 2 times the sine 00:13:48.349 --> 00:13:59.855 of 60 degrees. This is a good chance to review one of our exact value angles. Sine of 60 degrees is the 00:13:59.857 --> 00:14:08.597 square root of 3 over 2. So what am I asking here? I'm asking, "What angle has the sine square root of 3?" 00:14:08.599 --> 00:14:15.909 No angle has the sine square root of 3. Remember sine is always between 1 and negative 1, so this angle 00:14:15.911 --> 00:14:29.159 does not exist. It does not exist and so there are zero triangles given this particular set of measurements. 00:14:30.908 --> 00:14:35.405 Now let's do an example with the ambiguous case that has two triangles, and I want to show you how you 00:14:35.407 --> 00:14:40.673 can discover that there are in fact going to be two triangles. Just from looking at this data you can't tell. 00:14:40.675 --> 00:14:44.787 We're going to continue just like we did in the past. The first thing we're going to do is we're going to 00:14:44.789 --> 00:14:54.785 find this first angle gamma. So we know, from the law of sines, that the sine of gamma is to 10 as the sine 00:14:54.787 --> 00:15:08.561 of 35 degrees is to 6. So we have that the sine of gamma (oops I wrote alpha, not gamma). The sine of gamma 00:15:08.563 --> 00:15:23.572 is 10 times the sine of 35 degrees divided by 6, so gamma is sine inverse of 10 times the sine 35 degrees 00:15:23.574 --> 00:15:30.583 divided by 6. So now we're going to find this first gamma. Again, you have to be careful when you're 00:15:30.585 --> 00:15:36.586 putting in your values. Make sure you put parentheses around that numerator. So we're going to have 00:15:36.588 --> 00:15:50.601 sine inverse of 10 times the sine of 35 divided by 6, and we get a value of 72.9 degrees. Let's make that 00:15:50.603 --> 00:16:02.600 approximate. Alright, 72.9 degrees. So we know that we have this triangle. Gamma is approximately 00:16:02.602 --> 00:16:11.124 72.9 degrees. Now remember there are two angles less than 180 degrees that have this same sine, so we're 00:16:11.126 --> 00:16:19.380 going to find, we'll call this one gamma one, so we're going to find gamma two. So gamma two is 180 degrees 00:16:19.382 --> 00:16:36.131 minus 72.9 degrees, and that's going to be 107.1 degrees. We'll just do one decimal place. So here's the question. 00:16:36.133 --> 00:16:49.159 Can we have two triangles? Well, if instead of 72.9 degrees I wrote in 107.1 degrees, the question is 00:16:49.161 --> 00:16:59.159 could that work? Yes, it can because 35 plus 107.1 degrees is less than 180 degrees. Now I know I have 00:16:59.161 --> 00:17:06.680 two triangles, so we're going to make another chart here with our data. So we'll put back here, we have that 00:17:06.682 --> 00:17:19.192 this is 72.9, and now this is going to be triangle one, and we're going to have triangle two. So we know that 00:17:19.194 --> 00:17:27.195 a is 6. I'm going to call that b one and this beta one and that gamma one just to keep myself straight so 00:17:27.197 --> 00:17:34.454 I don't get confused. So this will be b two. [Sides] a and c have the same values, so that doesn't change. 00:17:34.456 --> 00:17:43.204 Alpha is 35 degrees. This will be beta two, and this will be gamma two which we've already found. So that's going 00:17:43.206 --> 00:17:51.701 to be approximately equal to 107.1 degrees. Now we have our two triangles set up, and it's really not going 00:17:51.703 --> 00:17:58.478 to be very difficult to solve for these missing parts. We can find these two angles by subtracting from 180 00:17:58.480 --> 00:18:12.249 degrees. So we know that beta one is approximately 180 degrees minus 35 degrees minus 72.9 degrees. 00:18:12.251 --> 00:18:22.254 Alright, so we have 180 minus 35 minus 72.9, and that's going to be 72.1 degrees. They're kind of close, 00:18:22.254 --> 00:18:31.492 and we'll talk about what happens there in a minute. So then beta two is approximately 180 degrees 00:18:31.494 --> 00:18:47.500 minus 35 degrees minus 107.1 degrees, and let's see what that is. That's 37.9, so this is approximately 00:18:47.502 --> 00:18:57.789 37.9 degrees. We have found everything but these two sides. Now I want to show you something here, 00:18:57.791 --> 00:19:02.294 and this is kind of a way you can... You know there's always ways that you can check yourself and maybe 00:19:02.296 --> 00:19:07.047 not know exactly, but you can tell if you're kind of going in the right track. Remember we talked about the 00:19:07.049 --> 00:19:12.045 fact that the largest angle is opposite the largest side, and they kind of have these ratios. You'll notice that 00:19:12.047 --> 00:19:18.797 these two angles are really close in value, so if I do my work correctly, I would expect that these two 00:19:18.799 --> 00:19:26.317 sides should be really close in value, and you've got the same thing down here. This is 35. This is 37, not 00:19:26.319 --> 00:19:32.567 quite as close as they are, but that's going to kind of give you an idea that b two is going to be just a little 00:19:32.569 --> 00:19:38.334 bit bigger than 6, and this one is going to be just a little bit less than 10. So that will kind of help you when 00:19:38.336 --> 00:19:45.582 you're doing your calculations. Make sure you don't put something in wrong. So let's find b one. [Side] b one is 00:19:45.584 --> 00:19:54.344 to the sine of 72.1 degrees, now remember once again use what you're given not what you find, so as 00:19:54.346 --> 00:20:10.612 6 is to the sine of 35 degrees. So b one is 6 times the sine of 72.1 degrees divided by the sine of 35 degrees. 00:20:10.614 --> 00:20:24.378 Alright, so let's see what that is. 6 times the sine of 72.1 divided by the sine of 35, and that's 9.95, and that's 00:20:24.380 --> 00:20:32.383 what I expected. I wanted it to be close. Now we'll find b two. Again, we're going to use what we're given, 00:20:32.385 --> 00:20:50.397 not what we've found, so we set up that b two is to the sine of 37.9 degrees as 6 is to the sine of 35 degrees. 00:20:50.399 --> 00:21:02.913 So b two is 6 times the sine of 37.9 degrees divided by the sine of 35 degrees. Let's calculate that. 00:21:02.915 --> 00:21:17.412 6 times the sine of 37.9 divided by the sine of 35 is 6.43. Again, that's what I expected. I wanted those to 00:21:17.414 --> 00:21:21.666 be close because the angles are close, so that's just kind of a visual check you can give yourself. 00:21:21.668 --> 00:21:27.925 So now we have solved the law of sines case where we have two triangles.