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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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##### Alisa L.

University of Texas at Austin

##### Boris M.

University of North Carolina at Chapel Hill

##### Martha R.

Michigan State University

##### Kristen K.

University of Michigan - Ann Arbor

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Co. Since two Find the missing sides and angles of a Triangle. ABC, where Angle C is 35 degrees Side A is 11. Inside B is 10.5, so let's go ahead. Let's start by finding our first missing side, which is side see and Laws Will Cosign says that we're trying to find a missing side. We're going that side squared, plus the some of the other two sides squared, which would be a squared plus B squared in this case minus two times eight times be Those two sides were given times. The co sign of C will give me C squared and then we'll find that square root. So now that we have our formula, let's put in our numbers. So for C Square, we know a squared is 11. So have 11. Squared B is 10.5, so we have 10.5 squared minus two. Tom's 11 times 10.5 Tom's co side, and we know Angle C is 35 so let's go ahead and let's simplify this some more. So 11 squared is 121 10.5 squared is 110.25 minus Let's go ahead. Multiply two times 11 times 10.5, which is 231 cosign of 35 degrees. So when I simplify that even more 221 plus 110.25 is to 31.25 minus 2. 31. I apologize. When I multiplied to 31 times Coast sign of 35 degrees. I get 189.2. So that means that C squared is approximately 42.5 And so to find the square root Ah, 42.5. That means that C is approximately 6.5. So are missing Side is approximately 6.5. So now that we have that, we can go ahead and use law of signs to find one of our missing angles. So let's use the law. A sign that says Sign of C, divided by side See, would equal the son of a divided by a. So for this one, we know that C is 35. So we're gonna use sign of 35 degrees over 65 which was side see well equal sign of a, which is what we don't know. Over were given side A, which is 11. Thank you. Yeah. So now that we know that we're gonna cross multiply, so we'll have 11 sign of 35 with equal 6.5 sign A. I'm gonna go ahead and divide by 6.5 on both sides because I'm gonna try to find sign of a so 6 11 site sign. 35 is going to give me a 6.309 and we know we're gonna divide that by 6.5 and now equal San'a. So that that is 0.971 will equal sign A. So when I put that in my calculator and I do that reverse operation that tells me that angle A is approximately 76 degrees. So now that we know angle a and we know angle See, we confined angle B. We know that all three angles added together. Well, give me 180 so we can use that information to find our final angle. We know angle A is 76. We don't know B and we know C is 35. So when I add 76 35 I get 111 plus B well equal 1 80. And when I subtract, I confined that angle. B is 69. Do Greece.

Liberty University

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Graphing Trigonometry Functions

##### Alisa L.

University of Texas at Austin

##### Boris M.

University of North Carolina at Chapel Hill

##### Martha R.

Michigan State University

##### Kristen K.

University of Michigan - Ann Arbor