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Lawsof Cosines - Example 3

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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Let's look at Triangle ABC, where Side A is 21.5, Bs 13 and C is 38 degrees. We're gonna use law co signs to solve for this triangle, So let's start with trying to find side to see. We know that the law cosign says that C squared will equal a square plus B squared minus +28 times be cosign of C. So let's plug in our numbers. A squared is 21.5, so we'll square that B squared B is 13 squared minus two times 21.5 times 13 co sign, and we already know the angle C is 38 degrees. So let's go ahead and solve this. So we have C squared ISS 462 0.25 plus 169 minus 559 coasts on 38 so we can add and then are multiplying right here. So for 62.25 plus 1, 69 is 631.25 and when I multiply 5 59 times coastline of 38 I get 440.498 So I can go ahead and I can subtract that. Yeah, to where I have that C squared is 190 degree, 190.752. So I'm gonna find the square root of that. So that tells me that C is approximately 13.8. So now that I know that I can use my sigh lol sods Which says that son of a divided by a what? Equal sign A C divided by C we don't know Sign of a So we'll keep it. Son of a We know Side A is 21.5. We know Sign C is going to sign 38 degrees and C We just found this 13.8. So I'm gonna cross multiply. So we're gonna have sign of a times 13.8 like that in front will equal 21.5 son of 38. I'm gonna go ahead and divide by 13.8 because I know I'm going to be solving for sine of a so Sanna Bay will equal When I multiplied 21.5 times Sign of 38 I'm gonna get 13.237 And that will be divided by 13.8. So that means that son of a is approximately 0.959 And when I do that reverse operation in my calculator angle A is going to be approximately 74 degrees. So now let's find angle B. We know, angle a and we know angle, See? So we're just gonna use the angles of a triangle formula to solve for bay A. We just said was 74 degrees, we don't know B and we knew See, it was 38 because we were given that amount and we're gonna add those two together. So 74 plus 38 will give me 112 plus B will equal 1 80. And when I subtract 1 12 from both sides, I find that B is going to give me 68 degrees. So angle B is 68 degrees