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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 relate the sides of a triangle (or a polygon with three or more vertices) in Euclidean plane geometry. The law of cosines was known to ancient Indian and Babylonian mathematicians. The law of cosines was also known to ancient Chinese mathematicians, but the name of the law of cosines was given by European mathematicians.

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AH
##### Ahyeon H.

University of California, Berkeley

##### Boris M.

University of North Carolina at Chapel Hill

##### Shubham P.

University of Washington

##### Martha R.

Michigan State University

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when it comes to triangles. We can use this when we're given a triangle with an angle and two sides. For example. Here we have the Triangle ABC and were given angle see and were given side be and were also given side A were not given angle a angle B or C, but we think this is enough information that we can use Law of Cozzens on the law co sign says. Whichever side we're looking for, it's square will be the some of the other two sides squared minus to Tom's each of the sides then cosign of that opposite ankle. So when we look at this one were given angle see And we're looking We're gonna start off by looking for side See So this tells me that the law coastline we're gonna use a C squared minus a square plus B squared minus two a b co sign of C So we're gonna use C squared equals a squared plus B squared minus two a. C. I'm sorry, baby co sign of C So let's go ahead and let's plug in our values for that one. So we're gonna have C squared equals 18 squared, plus 24 squared minus two times 18 times 24 coasts on an angle, see is 57 degrees so we can simplify that 18 squared is 324 24 squared is 576. If I multiply negative two times 18 times 24 I get 864 times co sign of 57 degrees. Most simplify that even Mawr. And I'm gonna get 900 minus when I 864 times co sign of 57 is going to be approximately 470 0.568 And so this tells me that C squared will equal approximately 429 0.4. Now we're going to be looking we've got C Square. We want to find what see is approximately. So what we need to do is we need to find the square root of both of these. And, of course, the square root of C squared us. What is C? And the square root of 429.4 is approximately 20.7. So my side length for C is 27. Now that we have that, we can go ahead and we can solve for angle A and ankle be And we're gonna do this by going back to our laws of sun We know a sign we know, see? So we're gonna start there, we're gonna set up our proportion So we're gonna say sign of C Divided by the length of sazi. Yeah, is equal to let's just let's start with angle a sign of a divided by Assad A So we know that angle. See ISS 57 degrees And we've already found that side See is approximately 20.7. We don't know ankle A but we do know that Side A is 18. So now that we know that we can cross multiply, we can multiply 20.7 Son of a We're equal 18 Sign of 57. Since we're solving for sine of A, we're gonna divide by 20.7 on both sides. So let's simplify this some more. So we have son of a will equal. This is going to be approximately 15 0.96 over 20.7. So this tells me that sign of A is approximately 0.7 to 9. So now that I no sign of a we can actually find angle A And what we're gonna do is we're gonna do that reverse operation in our calculator. So you're gonna make sure you use that button right there, which will be above the sun button, usually have to do a second and then this, but and that's gonna be approximately 47 degrees. So we're gonna say the measurement of angle A is approximately 47 degrees. So now we know side see is approximately 20.7. Angle A is approximately 47. So we confined angle B and we're gonna find angle. Be just using our two angles so we know that angle A. It's 47 degrees and we know ankle. See, it's 57. We also know that ankle a plus ankle B plus ankle C is gonna give me a total of 180 degrees. So since a it's 47 degrees and CS 57 degrees, we can easily find Ankle Bay. So we must add 47 57 is 104. And when we subtract 104 from 1 80 that tells me that measurement of Angle B is 76 degrees

Liberty University

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AH
##### Ahyeon H.

University of California, Berkeley

##### Boris M.

University of North Carolina at Chapel Hill

##### Shubham P.

University of Washington

##### Martha R.

Michigan State University