 In this video, we provide the solution to question number 15 for practice exam number 3 for math 1060 We have to solve the triangle ABC where we know angle a is 30 degrees We know the side length a equals 1 centimeter and we know the side length b equals the square of 2 centimeters Exact answers are really great here. We don't need to know the fact square of 2 is like 1.4 or whatever This is actually a blessing to have the square of 2 right here Notice that we have an angle opposite side a os so I'm gonna use the law of science to find out the medicine information But I should caution you right we have an angle Let's see. We have a side. We have a side. We have an angle. That's the ssa. This is the ambiguous case So that's why the instructions say there could be multiple solutions There actually might not be any solutions. We have to check for that So the first thing we're gonna do is we're gonna look for sign of B again using law of science So sign of B over little b is equal to sign of a over little a This tells us that sign of B is equal to B over a times sign of a Or in other words, we get little b is the square of 2 we get little a is 1 and then we get sign of 30 degrees notice that sign of 30 degrees Is the same thing as one half so you get the square of 2 over 2? When does that happen? Well, this is something at the check remember when it comes to when it comes to sign sign is always bounded Not sim sign is always bounded above by 1 and below by negative 1 right? So you have to check to make sure this ratio is between 1 and negative 1 can't be bigger than 1 can't be less than negative 1 If it was that means there's no solution here now root 2 over 2 We're familiar with this this happens when B is 45 degrees. Well, that's if B is an acute angle We have to also check if B was an obtuse angle, right? What if it just references to 45 degrees but in the second quadrant? So that's 180 take away 45, which is 135 degrees We have to consider both possibilities. There's the possibility that B is 45 degrees There's the possibility that B is 135 degrees now when B is acute If you got past this marker, you're gonna be fine for the rest of it This one will definitely give us a triangle. So if B is 45 degrees Then that means that C is Equal to 180 degrees take away 30 degrees take away 45 degrees like so for which then 30 and 45 come together to give you 75 degrees and if you take 75 degrees away from 180 degrees you get C is 105 degrees so we're gonna label these things so C is 105 B is 45 now what we have to do is we need to solve for little C And we can use the law of we can use the law of signs to do that one So little C over sine of C is equal to little a over sine of a I'm gonna use a because that's the original AOS I had here solving for little C we get little C is equal to a sign of C over sine of a Like so little a was a one We have sine of a hundred and five degrees like so and then we have sine of 30 degrees that we have to compute here Right now sine of 30 degrees you probably remember that one That's gonna just be that's just a times one right there We don't even need the times one anymore a sign of 30 degrees That's just gonna be a one half like so sign of 105 degrees You could just plug this into your calculator and you get an approximate solution, which is acceptable But notice that 105 degrees it actually references 75 degrees because after all 180 take away 75 is 105 so if you're in the second quadrant your reference angle is actually your supplement So this is anything a sign of 75 degrees. Why is that significant? Well sign of 75 degrees You might recall is root 6 plus root 2 over 4 2 goes into 4 2 times so we get the answer of root 6 plus root 2 Over 2 as the final answer in this case the length here of course would be in centimeters Like so if you want an approximate solution, that's fine You maybe just did that on your calculator and your approximate solution, of course would be 1.932 centimeters to 3 decimal places So that's one possibility the other possibility is what if you actually were 135 degrees? We have to consider both so if B was equal to 135 That would then suggest then that C is equal to 180. Let me scooch this over a little bit It's equal to 180 minus 30 degrees. That's still a minus 135, which is now B So that would tell us that C is equal to 180 Take away 165 degrees like so notice 30 plus 135 is 165. That's not bigger than 180 So this is gonna be positive 180 take away 165 is 15 degrees So it turns out there is a second possibility You wouldn't have a you wouldn't have a second possibility if the sum of A and B turned out to be bigger than 180 Thus C is negative So then we see that little C by the law of signs It can be a times sine of C over sine of a just like we had before little a is still one We now have sine of 15 degrees Over sine of a's a is still gonna be 30 degrees. So you're gonna get a one-half in the bottom again We could again this so this is equal to two sine of 15 degrees You could use your calculator to approximate this But I'm gonna put the exact value because we know sine of 15 degrees like sine of 75 This is gonna be root 6 minus root 2 over over 4 But then the 2 and the 4 simplify to give us root 6 minus root 2 over 2 and this will be in centimeters This is the exact value the exact value is preferred But it is not necessary if you just put this in your calculator And you got 0.518 centimeters that also would be considered correct and therefore we found two solutions here Let me zoom out so you can see all of them We found two solutions to this one two triangles that work one where B was 45 and C is 105 and one where B was 135 and C was 15