Solve each triangle ABC.

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Video transcript

Welcome back. Everyone in this problem. We want to find the missing silence and angle of the triangle. ABC. For our answer choices, A says that angle B is 77 degrees. Side A is 39 centimeters and side C is 5151 centimeters. Approximately B says angle B is 87 degrees A is approximately 39 centimeters and C is approximately 51 centimeters. C says that B is 77 degrees A is approximately 51 centimeters and C is approximately 39 centimeters. And the D says B is 87 de angle B is 87 degrees. Side A is approximately 51 centimeters and side C is approximately 39 centimeters. No, if we're going to figure out the missing side lengths and anger of the triangle. ABC, let's first ensure we understand our naming convention here. So when we're talking about triangles, if we name this anger A, then the side opposite to anger A will be labeled as common A. Likewise the side opposite to anger B will be labeled as common B. So B is 57 centimeters long and therefore opposite to the C is side C and we're trying to figure out B and sides A and C. So far notice that we have two, we know two angles in our triangle angles A and C. So we can use that to figure out angle B and then see where we can go from there. So let's start by doing that. No solving for be, we know that in a triangle, the summer anger in a triangle sum to 180 degrees. So since angles in a triangle, let me write that thought come to 180 degrees, that means I gonna be OK? Or to find angle B, we can subtract the other two angles from 180 degrees. So angle B is going to be equal to 180 degrees minus 42 degrees plus 61 degrees. OK? No 42 and 61. Uh that's 103 and 180 minus 103 equals 77. So angle B is 77 degrees. Let's put that on our diagram now that we have a value for angle B. Let's see if we can solve for sides A and C. What do we know in our triangle? Well notice in our triangle here, angle B is opposite to our side B which is 57 centimeters. We know the angle under the side. So we have a pair of angles and sides in our other angles. We don't know their side, but we know the angle OK, the value of the angle. So with an unknown angle, sorry, an unknown side and known angle, we can use the S law to figure out that unknown side. Recall that by the sign law, OK. The same law tells us that for any triangle, the sign of an angle or sorry, the ratio of the sign of an angle to its opposite side is the same throughout the triangle. In other words, the side A, OK. The side A two, the sine of its opposite side opposite an A is going to be the same as B to the sine of Anger B which is going to be the same as C to the sine of C. So we're saying since we know side B and B, we can use that to figure out the lengths of side A and the length of side C let's start with side A. So something for A, OK. Then we can say that S A to the sign of angle A which is 42 degrees is going to be equal to the length of side B which is 57 centimeters two. The sign of angle be which is 77 degrees. OK? Since that's the case, now we can solve for A because that means A is if we multiply both sides by the sine of 42 degrees A is going to be equal to 57 sine, 42 divided by the sine of 77 degrees. So to solve for a, we can put this value into our calculator. Now, when you do that, OK, when you put this value into your calculator, you should get approximately or you should get a value which is equal to 39.143694 93 centimeters. OK. Which when we round that to the nearest centimeter, like our answers, you should get it to be approximately equal to 39 centimeters. So that's the value of a. Likewise, we said we can use the S law to figure out the value of side C OK? Because no, for side C again, using the sine law, let me put this in blue, it tells us then that side C to angle C OK? Which is 61 degrees. The sine of Angus C sorry is going to be equal to uh we could still use B the side B 57 centimeters OK? Two, the sine of 77 degrees. And not to solve for C we can multiply both sides by the sine of 61 degrees. So you're going to get C to be equal to 57 sine, 61 divided by the sine of 77 degrees. Again, when we take this value and we put it into our calculator and round it to the nearest centimeter like the rest of our answers, we should get C to be approximately equal to approximately equal to 51 centimeters. So what does this mean? Well, side A is approximately 39 centimeters. Side C is approximately 51 centimeters and angle B is 77 degrees A is our correct answer. Thanks a lot for watching everyone. I hope this video helped.