In Exercises 1–8, solve each triangle. Round lengths of sides to the nearest tenth and angle measures to the nearest degree.

Video transcript

Hey, everyone. Welcome back in this problem. We're asked to find the missing angles of the triangle and to express them to the nearest degree. So we're given a triangle ABC, we have side A is equal to 13, side B is equal to 16 and side C is equal to seven. Now, we have four answer choices A through D each containing different values for the three angles. And we're gonna come back to those as we work through this probe. So we're given three sides, we're looking for angles. So the first thing we're gonna do is use the cosine law and, and we're called the cosine law or the law of cosines. And we can write it as C squared is equal to A squared plus B squared minus two multiplied by a multiplied by B multiplied by cosine of angle C. Hm. Now we can rearrange this, we can write this in terms of side A or side B as well. But we're gonna start with side C. OK. And we could have used any side here because we know all three side points. OK. So this is gonna find us angle C, we know side C side A and side B substituting in what we know we have seven squared is equal to 13 squared minus oops plus 16 squared minus two, multiplied by 13, multiplied by 16, multiplied by cosine of C simplify, we have 49 is equal to plus 256 minus 416 cosine angle C if we subtract our 169 plus 256 to move them to the left hand side, we have negative 376 is equal to negative 416 multiplied by cosine of Anglesey, which tells us that cosine of angle C must be equal to negative 376 divided by negative 416. And so angle C must be equal to about 25.33 degrees. All right. So we found our first angle, we are asked to round to the nearest degree. OK. So for Anglesey, we found that it's about 25 degrees. If we look at our answer choices, option A has an angle C is 23 degrees. That's not correct. So we can eliminate it. Option B has that it is equal to 20 degrees again, not correct. Eliminate it. And we're left with either option C or option D. Now moving on to the next piece, we're gonna repeat the same thing. We're gonna use our law of cosigns. But now we're gonna try to find angle A. So let's give ourselves some more room. We're gonna repeat it. But with angle A, so now our equation is gonna be written that side A squared equal to B squared plus C squared minus two, multiplied by B multiplied by C multiplied by cosine of angle. Substituting in our values 13 squared is equal to squared plus seven squared minus two, multiplied by 16, multiplied by seven, multiplied by cosine of a simplifying 169 is equal to 256 plus 49 minus 224 cosine of a subtracting 256 plus 49 from both sides, we have negative 136 is equal to negative 224 multiplied by cosine of A. That tells us that cosine of A is going to be equal to negative 136 divided by negative 224. And if we take the inverse cosine, we get that A is about 52. degrees. All right. So we found our second angle. We're gonna round this to the nearest degree and that's going to give us degrees. If we look at our answer choices, option C has that A is equal to 53 degrees. Option D has that it's equal to 100 and two degrees. So it's looking like C is gonna be the correct answer. But we can find angle B as well. Just to be sure, just to double check, we haven't made any issue uh any mistakes with their calculations. Now, for angle, we could do a couple of different things. We could use our law of cosines. Again, we could also use our law of signs because we now have some angles and corresponding sides. Or we can use the fact that the sum of the angles A plus B plus C in the triangle will add up to 180 degrees. This is the simplest way to do it. So let's go ahead and do it. This way this gives us 52.62 degrees was B plus 25.33 degrees is equal to 180. If we subtract 52.62 and 25.33 from both sides, we get that B is about 102. degrees. And again, rounding to the nearest degree is going to give us 102 degrees. And so the correct answer is, in fact, option C A angle A is 53 degrees. Angle B is 102 degrees and angle C is degrees. And one thing I wanna note before we go here, you can kind of double check that it makes sense. The smallest side should always correspond to the smallest angle and same for the largest side. So the smallest side was C and we found that the smallest angle was C as well. The largest side was B then that's our largest angle. So that answer does make sense. All right. That's it for this one. Thanks everyone for watching. See you in the next video.