 In this video, we provide the solution to question number 12 for the practice exam number three for math 1060 We're given a triangle ABC and we're asked to solve for that triangle The information we know is that the angle measure of a is 45 degrees The length of side B is one inch and the length of side C is the square root of two inches Which is about 1.4, but it's better to have the exact value there so notice that we have a Side angle side situation, so the law of cosines of the very appropriate tool for us We know angle a we can use the law of cosines to find the length of side length a right there So that's how we're going to proceed So the law of cosines tells us that a squared plus B squared or sorry a squared equals B squared plus C squared minus 2 BC cosine of a so we don't know what little a is so we're gonna fill out the right-hand side here Little B is one so we get one squared little C is the square root of two. We're gonna square that which is helpful We're gonna get two times one times the square root of two then times that by the cosine of 45 degrees for which one squared is one the square root of two squared is two Which is why it's better to have the exact value there two times one times square of two is two times the square root of two And we do are allowed to calculate here, but 45 degrees. That's one of our special angles That's gonna be root two over two For which then we're gonna continue to simplify this thing one plus two is three we're gonna have a Square root of two times the square root of two that's a two and then these two's cancel out like so So you end up with a minus two so in the end we end up with a one for that So a squared is equal to one taking the square root We get that little a is equal to positive one inch like so so that we see that's gonna equal one right there That helps us out a lot here because if this side is one and this side is one that actually means we have any We have an assosceles triangle The this Angle opposite side is corresponding to this angle opposite side they're equal that actually tells us very quickly that we have 45 degrees If you're not satisfied with that quick use of the assosceles triangle theorem We can use the law of sines to help us out here sine b over little b is equal to sine a over little a Right this would then of course tell us that sine of b is equal to b over a times sine of a Which b over a we both know those are one one over one times sine of a That is 45 degrees for which that would simplify to give us root two over two When is sine of b equal to root two over two you can consult your calculator But you're gonna end up of course with b equals 45 degrees again like so The other temptation of course is to do like in the second quadrant you're gonna get a hundred and 35 degrees which of course would leave that angle c is zero so that one doesn't work So we get that B is 45 degrees, but like I said, I recognize with a sausage triangle is able to do that really quickly That's not a necessary observation, but it does simplify the calculation dramatically there Once you get angle a and b whether you use the law of sines or the assosceles triangle theorem That doesn't matter once you get a and b to find the measure of angle c We're just going to of course take a hundred and eighty degrees Subtract from the angles we know so we have 45 degrees 45 degrees which combined is 90 degrees So a hundred and eighty degrees minus 90 degrees so we get that C itself is 90 degrees so it turns out that in hindsight we see that this triangle was actually a 40 40 45 45 90 triangle which was pretty cool not necessary. We found this using the law of cosines and law of sines mind you But we found all this information and it turned out to be pretty cool at the end Don't worry about the diagram these diagrams are not drawn the scales The fact doesn't look like a right triangle is actually important because we didn't want that to suggest That was the right answer. We found it through this trigometric formula of the law of cosines and sines