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# Solving for a side with the law of cosines

CCSS.Math:

## Video transcript

let's say that I've got a triangle and this is this is the side has length B which is equal to 12 12 units or whatever whatever units of measurement we're using let's say that this side right over here this side right over here has length C and that happens to be equal to 9 and that we want to figure out we want to figure out the length we want to figure out the length of this side and this side has length a so we need to figure out what a is going to be equal to now we won't be able to figure this out unless we also know the angle here because these things could be you could bring the side this the blue side and the green side close together then a would be small but if this angle was larger than a would be larger so we need to know what this angle is as well so let's say that we know that this angle which we will call theta is equal to 87 degrees so how can we figure out a and I encourage you to pause this and try this on your own well lucky for us we have the law of cosines which gives us a way for determining a third side if we know two of the sides and the angle between them the law of cosines tells us that a squared is going to be equal to B squared plus C squared now if we were dealing with a pure right triangle if this was 90 degrees then a would be the hypotenuse and we would be done this would be the Pythagorean theorem but the law of cosines gives us an adjustment to the Pythagorean theorem so that we can do this for any arbitrary angle so the law of cosines tells us going to be a squared is going to be B squared plus C squared minus 2 times BC 2 times B C times the cosine times the cosine of theta times the cosine of theta and this theta is the angle that is that opens up to the side that we care about so we can use theta because we're looking for a if they gave us another angle right over here that's not the angle that we would use we care about the angle that opens up into the side that we are going to four so now let's solve for a because we know what BC and theta actually are so a squared a squared is going to be equal to B squared so it's going to be equal to 144 plus C squared which is 81 so plus 81 minus 2 times B times C so that's minus 2 I'll just write it out minus 2 times 12 times 9 times 9 times the cosine cosine of 87 degrees cosine of 87 degrees and this is going to be equal to this is going to be equal to let's see this is 225 225 minus let's see 12 times 9 is 108 108 times 2 is 216 minus 216 times the cosine of 87 degrees 87 degrees now let's get our calculator out in order to approximate this and remember this is a squared actually before I get my calculator out let's just solve for a so a is just going to be the square root of this so a is going to be equal to the square root of all of this business which I can just copy and paste it's going to be equal to the square root of that so let me copy and paste it so a is going to be equal to the square root of that which we can now use the calculator to figure out so let me increase this radical a little bit so that we make sure we're taking the square root of this whole thing so let me get my calculator out so I want to find the square root of 220 actually before I do that let me just make sure I'm in degree mode and I am in degree mode because we're obviously finding the we're evaluating a trig function in degrees here so that's fine so let me exit so it's going to be 225 minus 216 times cosine cosine of 87 degrees not 88 degrees 87 degrees and we deserve a drum roll now this is going to be fourteen point six one or is it fourteen point six one eight if say we wanted to round to the nearest hands just to get an approximation it would be approximately fourteen point six so a is approximately equal to fourteen point six however whatever units were using long