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# In mathematics, the law of cosines, also known as the cosine rule or the cosine formula, is a formula used to calculate the cosine of an angle in a triangle, given the lengths of the sides of the triangle and the angle itself. It is one of the basic facts in Euclidean geometry.

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##### Boris M.

University of North Carolina at Chapel Hill

##### Martha R.

Michigan State University

##### Heather Z.

Oregon State University

##### Kristen K.

University of Michigan - Ann Arbor

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### Video Transcript

coastlines to find the missing angles of a triangle whose sides are 345 648 and 442 ah units in length. So the law cosign says that Side a squared would equal the sum of B squared and C squared minus two times B c times cosign of angle A. Now, even though we don't know in other examples, we've known at least one angle this time we don't. So we're gonna be finding just our angles. But we're still using Walcott since so now that we have our formula and we know our sides, let's plug those in. So a is 345 square equal 648 squared plus 442 squared minus two times 6, 48 times 442 cosign of a. So let's square these, we're gonna have some really large numbers, and that's okay. So for 345 squared, we're gonna have 119,000 25 will equal 419,904 plus 195,000 364 minus. When I multiply. I'm gonna have multiplied two times. 6, 48 times 442. I'm gonna get 572,832 cosign of a Okay, so now let's simplify this even more. Let's add 419,904 plus 195,364. So I have 119,000 25 will equal. And when I add those two numbers up, I'm going to get 615,268 minus 572,832 cosign of a So I'm gonna go ahead and subtract 16, 615,000, 268 from both sides. And when I do, I'm gonna get negative. 496,000 243 equals negative 572,000 832. Coast son of a So we'll divide both sides by negative 572,832. And when I get is 0.866 is approximately of cosign a and so I do the reverse operation in my calculator and I see that angle A It's gonna be approximately 30 degrees, so angle a will be approximately 30 degrees. So now that we use law co sides to find angle A let's use law Coast, this will use law signs to find another angle. And this time we're gonna look for angle seat. So Lawson says, sign of a divided by side a week will sign of sea divided by C So we have sign of 30 which we just failed over 345 Will equal son of C which we don't know over 442 we're gonna cross multiply and when I dio I'm gonna get 345 times Sanusi equals 442 Sign of A We're gonna divide both sides by 345 because we're trying to find sign of C. So we have now signed A C will equal 442 times. Son of A is going to give me 221 divided by 345 which means that sign of C is gonna be approximately 0.641 So when I go ahead and I do the reverse operation for a sign of See, I see that see can be 40 degrees or approximately 40 degrees. So now that we know that we can easily find angle, be because we know that angle a plus angle B plus angle see is 180 degrees. We know angle A is 30 degrees. We don't know be and we know see, it's 40 degrees well 30 and 40 70 so 70 plus B will have to equal 1 80. So when I subtract 1 80 minus 70 I can see that Angle B is 110 degrees.

Liberty University

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Graphing Trigonometry Functions

##### Boris M.

University of North Carolina at Chapel Hill

##### Martha R.

Michigan State University

##### Heather Z.

Oregon State University

##### Kristen K.

University of Michigan - Ann Arbor