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# Solving for an angle with the law of sines

## Video transcript

say you're out flying kites with a friend and right at this moment you're 40 meters away from your friend and you know that the the length of the kites string is 30 meters and you measure the angle between the kite and the ground where you're standing and you see that it's a 40 40 degree angle and what you're curious about is whether you can use your powers of trigonometry to figure out the angle between the string and the ground and I encourage you to pause the video now and figure out if you can do that using just the information that you have so whenever I see a I guess a non right triangle where I'm trying to figure out some lengths of sides or some lengths of angles I immediately think do I maybe the law of cosine might be useful or the law of cosines might be useful and so let's think about which one could be useful in this case law of cosines and I'll just rewrite them here the law of cosines is C squared is equal to a squared plus B squared minus 2 a B cosine of theta and so what it's doing is it's relating it's relating three sides of a triangle so a B C to an angle so for example if I knew two sides and the angle in between them I could figure out the third side or if I know all three sides then I could figure out this angle but that's not the situation that we have over here we're trying to figure out this question mark we're trying to figure out this question mark and we don't know three of the sides we're trying to figure out an angle but we don't know through the side so the law of cosines doesn't seem at least in an obvious way that's going to help me I could also try to find this angle but once again we don't know all three sides to be able to solve for the angle so maybe law of sines could be useful so the law of sines and the law of sines so let's say that this is this is the measure of this angle is a measure this angle is lowercase B measure this angle is lowercase C length of this side is capital C length of this side is capital a length of the side is capital B law of science tells us the ratio between the sine of each of these angles and the length of the opposite side is so sign of lowercase a over capital a is the same as the sign of lowercase B over capital B which is going to be the same as the sign of lowercase C over capital C so let's see if we could leverage that somehow right over here so we know we know this angle and the opposite side and the opposite side so we could write that ratio sine of 40 degrees over 30 let's see can we say that that's going to be equal to the sine of this angle over that well it would be but that we don't know either of these so that doesn't seem like it's going to help us but we do know this side and so maybe we could use the law of cosines to figure out this angle because if we know two angles of a triangle then we could figure out a third angle so let's do that so let's say that this angle right over here is Theta we know this distance right over here is forty meters so we could say that the sine of theta sine of theta over 40 this ratio is going to be the same as the sine of 40 over 30 and now we can just solve for theta multiplying both sides times 40 times 40 you're going to get let's see 40 divided by 30 is 4/3 for 4/3 sine of 40 degrees is equal to sine of theta is equal to sine of theta and now to solve for theta we just have to take the inverse sine of both sides so inverse sine of 4 over 3 sine of 40 degrees and put some parentheses here is equal to theta is equal to theta that will give us this angle then we can use that information of this information to figure out the angle that we really care about so let's get a calculator out and see if we can calculate it so let me just verify I am in degree mode very important all right now I'm going to take the inverse sine of 4/3 times sine of 40 degrees and that gets me I deserve a little bit of a drum roll now 58 well let's just rhyme it's well it's very close if we round to the nearest let's just so let's maintain our precision here so fifty eight point nine nine degrees roughly so this is this is approximately equal to fifty eight point nine nine degrees so if that is fifty eight point nine nine degrees what is this one well it's going to be 180 minus this angles measure minus that angles measure so let's calculate that so it's going to be 180 degrees minus this angle so minus 40 minus the angle we just figure it out and actually I could get all of our precision by just typing in second answer so that just says our previous answer so they get all that precision there and so I get 81 point zero one degree so if I want to round to the nearest let's say I round round to the nearest hundredth of a degree then I'd say 81 point zero one degrees so this this right over here is approximately 81 point zero one degrees and we're done