let's say side a is 10 and side b is 20 and angle c is 60 degrees go ahead and solve the triangle so first let's draw it so this is angle a b and c so angle c is 60 degrees side a is 10 side b is 20. so what we have is a side angle side triangle can we use law of sines to solve the triangle in order to use law of sines you need to have two of the same letter notice that we can't use it we have one of each different letter so in this case we need to use the law of cosines if you try to use the law of sines you're going to miss something for example let's say if we try to use a over sine a which is equal to b over sine b it's not going to work we have a and b but we don't have angle a nor do we know angle b and if we try to use b over sine b which is equal to c over sine c we're still missing angle b and we're missing side c so whenever you have all different letters you cannot use the law of sines to solve it however there's something else that we can use and that is the law of cosines so here's the formula that you need c squared is equal to a squared plus b squared minus 2 a b cosine of angle c now you can change it up and write two other forms a squared is equal to b squared plus c squared minus two b c cosine of angle a or b squared is equal to a squared plus c squared minus 2 ac cosine of angle b you can use any one of these three forms but i'm going to use the first one because we have everything to use that formula we have side a and b a is 10 b is 20. and we have angle c which is 60 degrees so we can use this to find side c 10 squared is 100 20 times 20 is 400 and 2 times 10 is 20 times another 20 that's 400 as well now cosine 60 is one half 100 plus 400 is 500 and half of 400 is 200 so c squared is equal to 300 so therefore c is the square root of 300 which is 10 root 3 or 17.32 so now that we have side c we could use the law of sines to figure out everything else so let's use the law of sines to find angle b c over sine c is equal to b over sine b side c is 17.32 and angle c is 30. b is 20. let's go ahead and find angle b so let's cross multiply so 20 times sine 60 that's 17.32 and that's equal to 17.32 sine b so if we divide both sides by 17.32 what that means is that sine b is equal to one so therefore b is the arc sine of 1 which is 90 degrees so there's only one answer here because if you do 180 minus 90 you're going to get 90 again now to find angle a that's going to be 180 minus 90 minus 60 which is going to be 30. and so that's how you can solve this particular triangle it turns out that it's a right triangle let's say that side a is seven side b is eight side c is nine use the law of cosines to solve the triangle so this time we have all three sides so what we have is a a side side side triangle so let's start with this formula c squared is equal to a squared plus b squared minus two a b cosine of angle c so first let's subtract both sides by a squared and b squared if we move it to the left side we're going to have c squared minus a squared minus b squared is equal to negative 2 a b cosine of c now let's divide both sides by negative 2 a b so this is the formula we're going to use cosine of angle c is equal to c squared minus a squared minus b squared divided by negative two a b c is nine a is seven and b is eight nine squared is 81 7 squared is 49 and 8 squared is 64. so 81 minus 49 minus 64. that's equal to negative 32 and then 2 times 7 times 8 that's 112. so negative 32 divided by negative 112 that's 2 over 7 which as a decimal is 0.2857 that's equal to cosine of angle c so angle c is going to be the arc cosine of that number 0.2857 so you should get about 73.3 degrees actually 0.4 for angle c now let's use the law of cosines to figure out everything else so let's start with this equation c over sine c is equal to a over sine a so c is nine angle c is 73.4 side a is seven now let's cross multiply 7 times sine 73.4 that's about 6.7 zero eight and that's equal to nine times sine a so six point seven zero eight divided by nine that's about point seven four five three so angle a is going to be the arc sine of that number so that's 48.2 degrees now let's find the other answer so a could be 180 minus 48.2 which is 131 but this answer is not possible because if we add this to the pre-existing angle that exceeds 180. so therefore there's only one possible solution one triangle that can be formed so a let's write the answer that's 48.2 degrees and now let's calculate angle b so b is going to be 180 minus 73.4 minus 48.2 and that's uh 58.4 degrees so if you have all three sides you need to use the law of cosines to find the first angle and then you can use the law of sines to find everything else so that's it for this lesson you