WEBVTT 00:00:07.607 --> 00:00:12.354 Sometimes when you're solving an oblique triangle, you run into a situation where the law of sines doesn't 00:00:12.356 --> 00:00:18.335 apply. The question is, "When does the law of sines not apply?" Well if you think about it, if you put your 00:00:18.337 --> 00:00:26.593 data down in an organized little chart like this, to use the law of sines you have to have a ratio. So what 00:00:26.595 --> 00:00:34.613 happens if I have side a, angle beta, and side c? I can't set up a ratio. There's no way to find side b. 00:00:34.615 --> 00:00:41.346 There's no way to find either one of these angles, so I don't have any options there. Or if I have all three sides, 00:00:41.348 --> 00:00:47.361 I'm not going to be able to use the law of sines. So these are the cases. This is the side-angle-side and the 00:00:47.363 --> 00:00:55.124 side-side-side case. When this occurs, you use the law of cosines. Now I have all three of these up here, 00:00:55.126 --> 00:01:02.629 or actually six of them. You have a squared is equal to b squared plus c squared minus 2 bc cosine alpha. 00:01:02.631 --> 00:01:12.130 Notice that this is a and this is alpha and that we have then alpha is cosine inverse b squared plus c squared 00:01:12.132 --> 00:01:19.627 minus a squared over 2 bc. As you study these you'll see the similarities between all of them. They're very 00:01:19.629 --> 00:01:24.402 similar. You just have some letters that are going to change. This is b squared, and this is a and c, and 00:01:24.404 --> 00:01:29.652 a and c and beta, and so on like that. Now, I want to warn you about one thing. When you use your calculator 00:01:29.654 --> 00:01:35.162 to find, and you're using cosine inverse, make sure you put parentheses around the numerator 00:01:35.164 --> 00:01:40.188 and the denominator and then around the whole expression. Otherwise it won't come out for you. 00:01:40.190 --> 00:01:44.000 So now we're going to look at an example using the law of cosines. 00:01:46.600 --> 00:01:54.365 Suppose that a is equal to 2, b is equal to 3, and angle gamma is equal to 95 degrees. If we look in this 00:01:54.367 --> 00:02:00.623 triangle up here and check off the information that we're given, we can see that this is the side-angle-side. 00:02:00.625 --> 00:02:07.637 It doesn't always have to be a and then beta and then c. It's the side-angle-side scenario. So we're going to 00:02:07.639 --> 00:02:19.641 use the law of cosines to find side c first. We know that c squared is equal to a squared plus b squared minus 00:02:19.643 --> 00:02:30.905 2 ab cosine of gamma. So c is going to be the square root of these values, and we're going to put these in. 00:02:30.907 --> 00:02:46.171 So that's 2 squared, which is 4, 3 squared, 9, minus 2 times 2 times 3 times the cosine of 95 degrees. 00:02:46.173 --> 00:02:54.683 So c is approximately equal to, alright we're going to find what this number is. Now the reason I'm not 00:02:54.685 --> 00:03:02.182 going to put a decimal approximation just yet is because when you're entering the values you want to 00:03:02.184 --> 00:03:07.688 make sure that when you switch these other values you want to use as little approximation as possible. 00:03:07.690 --> 00:03:19.439 We're going to go ahead and find that. That value is 4 plus 9 minus 4 times 3 times the cosine of 95, so we're 00:03:19.441 --> 00:03:32.952 going to say that this is 14.04587. We're just going to leave it like that. Now we can find an approximation 00:03:32.954 --> 00:03:41.718 for that and use that as a value for c over here. When I'm finding alpha or beta, whichever angle I find first, 00:03:41.720 --> 00:03:50.734 I'm going to use this number. The value of c is approximately 3.75. 00:03:50.736 --> 00:03:54.492 Now we're going to use the law of cosines to find angle alpha. 00:03:54.494 --> 00:04:09.001 So alpha is cosine inverse b squared plus c squared minus a squared divided by 2 times b times c. Now you 00:04:09.003 --> 00:04:15.000 can see why we left this in this notation because we can square it, get this number here, and then down here 00:04:15.002 --> 00:04:21.759 we'll use that square root notation. It's a lot to enter in your calculator, but you just have to be careful. So this 00:04:21.761 --> 00:04:34.011 is cosine inverse b squared, which is 3 squared, 9, plus c squared, which we'll use this approximation, minus 00:04:34.013 --> 00:04:54.045 a squared divided by 2 times b times c. Be careful when you put this in your calculator. You're going to 00:04:54.047 --> 00:04:58.560 have to have parentheses around your numerator and your denominator, and so this is going to take a lot 00:04:58.562 --> 00:05:18.573 of entering. So let's just do it. Just be real careful when we enter it. So that's going to be 14.0457 minus 4, 00:05:18.575 --> 00:05:42.845 (and that's 14.0457), so we get alpha is approximately 32.1 degrees. So now we can find beta by subtracting 00:05:42.847 --> 00:06:00.615 from 180. So we have 180 minus 32.1 minus 95, and so we get approximately 52.9 degrees. 00:06:00.617 --> 00:06:05.620 So now we've solved this triangle using the law of cosines. 00:06:07.500 --> 00:06:12.122 Now we're going to do an example solving a triangle when you're given all three sides. Again, 00:06:12.124 --> 00:06:18.380 we're going to use the law of cosines. What we're going to do is we're going to find the largest angle first. 00:06:18.382 --> 00:06:24.380 Sometimes when people use the law of cosines after they find the first angle they change to the law of sines. 00:06:24.382 --> 00:06:30.632 If you use that technique, then you definitely want to use the law of cosines to find the largest angle first. 00:06:30.634 --> 00:06:48.631 So we'll find angle gamma. Gamma is cosine inverse of a squared plus b squared minus c squared over 2 ab. 00:06:48.633 --> 00:07:00.420 Oops, that doesn't look like a gamma. It looks like an alpha. So that's cosine inverse a squared is 36, 00:07:00.422 --> 00:07:15.672 b squared is 25, c squared is 64, divided by 2 ab. a is 6, and b is 5. So we'll put that in our calculator, 00:07:15.674 --> 00:07:24.940 and find out what alpha is. So that's cosine. Again, make sure you put parentheses around the numerator 00:07:24.942 --> 00:07:37.448 and the denominator when you're putting this in your calculator, or you won't get the correct answer. 00:07:37.450 --> 00:07:51.725 So for alpha, excuse me, for gamma, we get approximately 92.9 degrees. Now, it doesn't really matter 00:07:51.757 --> 00:07:57.731 which of these other angles we find. I'm going to find alpha, and I'm going to use the law of cosines again. 00:07:57.733 --> 00:08:02.979 These are all exact numbers. They're easy to enter into your calculator, so that's not going to be too much 00:08:02.981 --> 00:08:14.995 to worry about. So alpha is going to be cosine inverse of b squared plus c squared minus a squared 00:08:14.997 --> 00:08:36.246 divided by 2 times b times c. So alpha is cosine inverse b squared, 25, plus 64 minus 36. 2 times b times c. 00:08:36.248 --> 00:08:50.990 So we'll discover what that one is. 00:08:50.992 --> 00:09:04.542 Alright, so we get alpha is approximately 48.5 degrees. Now we can find beta. We know the sum of the angles. 00:09:04.544 --> 00:09:17.308 That's going to be approximately. The sum of the angles is 180, so we'll just subtract there, and let's see what 00:09:17.310 --> 00:09:31.311 we get for that. So we get beta is approximately 38.6 degrees, so we've found these three angles. 00:09:31.313 --> 00:09:36.078 We found the first two using the law of cosines. The third one was just subtraction. 00:09:36.080 --> 00:09:40.590 So now we've solved a triangle with the side-side-side scenario.