WEBVTT
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Sometimes when you're solving an oblique triangle, you run into a situation where the law of sines doesn't
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apply. The question is, "When does the law of sines not apply?" Well if you think about it, if you put your
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data down in an organized little chart like this, to use the law of sines you have to have a ratio. So what
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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.
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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,
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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
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side-side-side case. When this occurs, you use the law of cosines. Now I have all three of these up here,
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or actually six of them. You have a squared is equal to b squared plus c squared minus 2 bc cosine alpha.
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Notice that this is a and this is alpha and that we have then alpha is cosine inverse b squared plus c squared
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minus a squared over 2 bc. As you study these you'll see the similarities between all of them. They're very
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similar. You just have some letters that are going to change. This is b squared, and this is a and c, and
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a and c and beta, and so on like that. Now, I want to warn you about one thing. When you use your calculator
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to find, and you're using cosine inverse, make sure you put parentheses around the numerator
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and the denominator and then around the whole expression. Otherwise it won't come out for you.
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So now we're going to look at an example using the law of cosines.
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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
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triangle up here and check off the information that we're given, we can see that this is the side-angle-side.
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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
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use the law of cosines to find side c first. We know that c squared is equal to a squared plus b squared minus
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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.
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So that's 2 squared, which is 4, 3 squared, 9, minus 2 times 2 times 3 times the cosine of 95 degrees.
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So c is approximately equal to, alright we're going to find what this number is. Now the reason I'm not
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going to put a decimal approximation just yet is because when you're entering the values you want to
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make sure that when you switch these other values you want to use as little approximation as possible.
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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
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going to say that this is 14.04587. We're just going to leave it like that. Now we can find an approximation
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for that and use that as a value for c over here. When I'm finding alpha or beta, whichever angle I find first,
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I'm going to use this number. The value of c is approximately 3.75.
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Now we're going to use the law of cosines to find angle alpha.
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So alpha is cosine inverse b squared plus c squared minus a squared divided by 2 times b times c. Now you
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can see why we left this in this notation because we can square it, get this number here, and then down here
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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
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is cosine inverse b squared, which is 3 squared, 9, plus c squared, which we'll use this approximation, minus
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a squared divided by 2 times b times c. Be careful when you put this in your calculator. You're going to
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have to have parentheses around your numerator and your denominator, and so this is going to take a lot
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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,
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(and that's 14.0457), so we get alpha is approximately 32.1 degrees. So now we can find beta by subtracting
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from 180. So we have 180 minus 32.1 minus 95, and so we get approximately 52.9 degrees.
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So now we've solved this triangle using the law of cosines.
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Now we're going to do an example solving a triangle when you're given all three sides. Again,
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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.
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Sometimes when people use the law of cosines after they find the first angle they change to the law of sines.
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If you use that technique, then you definitely want to use the law of cosines to find the largest angle first.
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So we'll find angle gamma. Gamma is cosine inverse of a squared plus b squared minus c squared over 2 ab.
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Oops, that doesn't look like a gamma. It looks like an alpha. So that's cosine inverse a squared is 36,
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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,
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and find out what alpha is. So that's cosine. Again, make sure you put parentheses around the numerator
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and the denominator when you're putting this in your calculator, or you won't get the correct answer.
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So for alpha, excuse me, for gamma, we get approximately 92.9 degrees. Now, it doesn't really matter
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which of these other angles we find. I'm going to find alpha, and I'm going to use the law of cosines again.
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These are all exact numbers. They're easy to enter into your calculator, so that's not going to be too much
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to worry about. So alpha is going to be cosine inverse of b squared plus c squared minus a squared
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divided by 2 times b times c. So alpha is cosine inverse b squared, 25, plus 64 minus 36. 2 times b times c.
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So we'll discover what that one is.
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Alright, so we get alpha is approximately 48.5 degrees. Now we can find beta. We know the sum of the angles.
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That's going to be approximately. The sum of the angles is 180, so we'll just subtract there, and let's see what
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we get for that. So we get beta is approximately 38.6 degrees, so we've found these three angles.
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We found the first two using the law of cosines. The third one was just subtraction.
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So now we've solved a triangle with the side-side-side scenario.