Solve each triangle. See Examples 2 and 3.

a ...

Question

Solve each triangle. See Examples 2 and 3.

a = 42.9 m, b = 37.6 m, c = 62.7 m

Accepted Answer

Welcome back. Everyone in this problem. We want to find the missing measurements of the triangle with sides, P being 64.5 yards. Q 52.3 yards and R 47.2 yards. For our answer choices. A says P is 80 degrees and 38 minutes. Q is 46 degrees and nine minutes and R is 53 degrees and 13 minutes. B says P is 46 degrees, 38 minutes, Q 53 degrees nine minutes and R is 80 degrees 13 minutes, C says P is 53 degrees, 38 minutes. Q is 80 degrees nine minutes and R is 46 degrees and 13 minutes. And D says P is 80 degrees 38 minutes, Q is 53 degrees nine minutes and R is 46 degrees 13 minutes. Now, to be able to figure out our missing measurements, let's try to draw all of the information that we have here. Let's try to put it on a triangle and our sketch doesn't have to be right. It doesn't have to be exact, but it's just to help us to get a sense of where all these measurements would be. So let's say that this is triangle P QR, OK. And for triangle Pqr, we know based on naming conventions for a triangle, if Anger P is here, then side P is opposite to Anger P. So this would be side P and based on the information we've given, we know that side P is 64.5 yards. Likewise, if this is an Q, then side Q is opposite to it, which is 52.3 yards. And if this is an R, then side R is opposite to it, which is 47.2 yards. So we are given the three sides of the triangle. And now we're supposed to try and figure out the three angles inside the triangle. Now, what do we know that relates to three sides of the triangle to any of the angles that are given? Well, recall that by the law of cosines, it tells us that if we have two sides and a known angle between them, we can use that to figure out the value of the side opposite to our angle. But that can go both ways. In other words, if we know the three sides, then we can use that to help us figure out the anger between them. In other words, by the law of course. And what we know is that P squared, side P is going to be equal to Q squared plus R squared minus two multiplied by Q multiplied by the cosine of P. So if we know PQ and R, we can rearrange our formula to solve for angle P like we would here. So let's try to transpose for Ango P and then substitute the values of PQ and R to solve. Now, we can make this expression two. QR cos P the subject of our equation which means if we add it to both sides and subtract P squared, two QR P is going to be equal to Q squared plus R squared minus P squared. Next, we can divide both sides by two. QR which means cos P equals Q squared plus R squared minus P squared divided by two QR which then means that the cosine, let me make some space to the left or to the right rather that means the value of P is going to be equal to the inverse cosine of our expression. And that makes sense because if we, if we know that a cosine of an anger equals an expression, then the inverse cosine of that expression will give us back our anger. So now we have an expression for P in terms of or for angle P in terms of sides, QR and P. So we can substitute those to solve for angle P because no, this means that P is going to be equal to the inverse cosine of Q squared. OK, which is the 52.3 plus R squared. Let me fix that here. OK. Plus R squared, which is 47.2 minus P squared, which is 64.5. So 64.5 squared all divided by two multiplied by Q which is 62.3 multiplied by R which is 47.2. OK. So that would be how we could find the value of angle P. Now, by doing that, OK, by doing that, by solving for P, we should find that angle P to the nearest minute is going to be approximately equal to 80 degrees and 38 minutes. So that's our value for angle P we've solved for P. Let's put that in our diagram. So here, OK, let me get rid of these. Now we know that P is 80.3 80 degrees. OK? And 38 minutes at angle peak. Now how could we use that to solve for any of the other angers? Well, again, we could use the law of cosines, but now we have an angle more specifically, we have a pair of angle and side. So because we have a pair of a known pair of angle and side, then we can use it to solve for any other angle in the triangle once we know the side that's opposite to it. And we do, we can use that to solve for R because we know that the side R is 47.2 yards and the angle R is unknown. So by the law of science, we should be able to solve for R. So let's go ahead and do that. And I'm going to do that in red. I should have put this in red actually. OK? Because no, OK. By the law of Science, by the law of science, recall that the law of science tells us that the ratio of the sine of an angle to its opposite side is the same throughout the triangle. So in other words, the sine of R two side R is going to be the same as the sine of angle P to side P. Now, we know that side R is uh you know R is 4 to 7.2 yards, OK? Angle P is 80 degrees and 38 minutes and side P is 64.5 yards. OK? We know all of those. Therefore, that means the sign of R then is going to be equal to 47.2 multiplied by the sine of 80 degrees, 38 minutes divided by 64.5. And again, we're in this situation, we know that the sine of an angle equals an expression. So to figure out that angle, we can find the inverse sine of our expression. In other words, this tells us then that R, OK, R is going to be equal to the inverse S of 47.2 multiplied by the sine of 80 degrees and 38 minutes divided by 64.5 and notice sulfur R, we can put that expression into our calculator. Now, when we do that, we should find that R to the nearest minute is approximately equal to 46 degrees and 13 minutes. So now we also have a value for Anger R. Let's put that value in our triangle. So now we know that R, I'm gonna put that in red as well. OK? Get rid of these here. Now we know that R is 46 degrees and 13 minutes. So now we have a value for R, we have a value for P. All we need to know is the sulfur value of Q. If we look at our diagram, we know two angles in the triangle. We want the third one. What do we know about angles in a triangle? Well, we know that angles in a triangle sum to 180 degrees. So if we can sum, if we can find here some at both of those angles and subtract them from 180 degrees, we'll get the value of Q. So solving for Q again, let's go here to the right of it. OK? Solving for a cue using the Angus some property of changes. OK? Then we now know that Q is going to be equal to 180 degrees minus the other two angles is minus P 80 degrees and 38 minutes minus R 46 degrees and 30 minutes. And now when we work that out, we find that anger Q equals 53 degrees and nine minutes. Ok. So from a working we have now found that that P is 80 degrees 38 minutes. R uh sorry, Q is 53 degrees nine minutes and R is 46 degrees 13 minutes. If we look back on our answer choices, the correct choice is answer choice. D Thanks for watching everyone. I hope this video helped.

Solve each triangle. See Examples 2 and 3.

a = 42.9 m, b = 37.6 m, c = 62.7 m

Key Concepts

Law of Cosines

Law of Sines

Triangle Properties

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