Solve each triangle. See Examples 2 and 3.

a ...

Question

Solve each triangle. See Examples 2 and 3.

a = 9.3 cm, b = 5.7 cm, c = 8.2 cm

Accepted Answer

Welcome back. Everyone in this problem, we want to find the missing measurements of the triangle with side P 3.5 m. Side Q 2.8 m and the side R being 5.1 m. For our answer choices, A P is 41 degrees, Q is 108 degrees and R is 31 degrees. B says it's 31 degrees, 41 degrees and 108 degrees respectively. C says it's 41 degrees, 31 degrees and 108 degrees respectively. And D says it's 108 degrees, 31 degrees and 41 degrees respectively. Now, if we're going to find the missing measurements of the triangle, let's try to just do a sketch to get an idea of where all of our measurements are. OK? And it doesn't have to be a perfect sketch. I just, we just want to get a sense of which sides are there and where their angles are and so on. So let's call this triangle triangle. P. QR, OK. P QR no four or triangle. We know that based on how we name triangle. If this is side P, then the side opposite. Or if this is angle P rather the side opposite to it would be called P, but denoted with a common in a common letter likewise opposite to angle Q is side Q and opposite to angle R would be side R. And so far, we know that side P is 3.5 m. So let me try to make some room for that. The P equals 3.5 m. Side Q is 2.8 m. OK? And the side R is 5.1 m. OK? And we're trying to find the missing Angus. Now, let's think about the information we have, we know the value of three sides. How could that help us to find any of the angles? Well, we're talking about a triangle here where we don't know if it's a right triangle. So we can't use any of the trigonometric ratios or the Pythagorean theorem. But there's one thing we know that relates all three sides of a triangle and that's the law of cosines because the law of cosines tells us that if we have two sides, if we know two sides and the anger between them, then we should be able to use that to find the side opposite to that anger. In this case, it's been reversed. No, we know two sides and the anger that's opposite to the anger between them. So we can be able to use that to find the angle. OK? Because what that means is that, let me put this in blue here by the law of Co science, by the law. Of course, science, we recall, the law tells us that R squared would be equal to P squared. OK plus Q squared minus two PQKR. Now we know R, we know P, we know Q and we can use that to figure out the value of R OK. Let's do some transposition first to make I the subject of the equation. Because no, that means two PQ QR OK is going to be equal to P squared plus Q squared minus R squared. OK. We added two P QR to both sides and subtracted R squared. That's how we got that. OK. And no, that means the cosine of R is going to be equal to P squared plus Q squared minus R squared all divided by two PQ. What does that mean? Well, no, that tells us then that R is going to be equal to the inverse cosine. So if we find the inverse cosine of both sides R will be the inverse cosine of P squared plus Q squared minus R squared divided by two PQ. And we know that because if the cosine of an angle equals this expression, then the inverse cosine of the expression will give us the angle. No, we can use that idea to solve for R OK. Because no, that tells us, then let me come over to the left hand side here on the right hand side of the page. No, this tells us, OK, this tells us that angle are, is going to be equal to the inverse cosine OK. Of P squared, which is 3.5 plus Q squared, which is 2.8 2.8 squared minus R squared, which is 5.1 squared. OK. Divided by two multiplied by P which is 3.5 multiplied by Q which is 2.8. OK. So that would be the value of R. So all we need to do now to figure out the value of R is to put this expression into our calculator. So let's go ahead and do that. Now, when you do that, OK, you should find that you get R to be equal to 107 OK? On your calculator, you should be seeing 107.5802206. But notice that all of our answers for our angles were to the nearest degree. So we can run the value of R then and say that R equals 108 degrees. That's, that's the value of R. OK? So, so far we know that angle R is 108 degrees. And that makes sense because the largest anger must be opposite to the longest side in a triangle like this. So R is 108 degrees. Let me get rid of this arrow as well. Now that we have the value of R, let's try to find another angle. Uh Let's say that we were trying to find the value of angle P, what could help us? Well, again, we could use the law of course as because we know the three sides. But notice here that no, we found the value of angle R, we have a pair of angle and side, a known pair of angle and side. And when we look at Anger P, we know the value of the side opposite to Anger P. When we are trying to find the value of this side, given two sides and an angle, then we can use the law of science because the law of science tells us that the sin of the ratio of the sine of an angle to its opposite side is the same throughout the entire triangle. So we could use that then to help us to solve for the value of P. Let me put that in red because I'm going to write that solution for P in red. So now by the law, by the Law of Science, eh by the law of Science, this tells us then that the sign of P to P is I think I need to let me just clean this up a bit here. So the sine of P to P is going to be equal to the sine of R to an R. Now, what do we know? Well, again, as we said, we know so far. And as we said, we know so far that um side P, the value of side P is 3.5 m, OK? We just found out to be 108 degrees and we know that R, the side R is 5.1 m long. So that means that the sign of P then is going to be equal to 3.5 multiplied by the sine of 108 degrees. OK? Divided by 5.1. And now again, the sine of an angle equals this expression. So to figure out what the angle is, we can use the inverse sine because no, that means P is going to be equal to the inverse sine of that expression 3.5 S 108 degrees divided by 5.1. And not to solve a four P, we can put that into our calculators. OK. Now, when we do that, we should find that angle P to the nearest degree is approximately equal to 41 degrees. OK. So that's our value for angle P, that's, that's what we know angle P is. So let's take that and put it back into our uh our triangle. OK. So we found R to be 108 degrees. And now we know that angle P an P is equal to 41 degrees. Finally, we need to solve for an Q. Now let's think about what we have. We have so much information here at this point, we know two of the angles inside the triangle. We just need to figure out the last one. What do we know about angles in a triangle? Well, we know that angles in a triangle sum to 180 degrees. Therefore, we can solve for Q because we know angles P and R. So let's use that property then. So here again, let me come a bit over to, over to the right here. They using the Anson property, OK? We know that P QR is going to be equal to 180 degrees. Therefore, the angle we're trying to find angle Q must be equal to 180 degrees minus angles P and R. So substituting the values of P and RQ is going to be 180 minus 108 degrees minus 41 degrees, which tells us that Q is going to be equal to 31 degrees. Therefore, angle P is 41 degrees. Angle Q is 31 degrees and angle R is 108 degrees. If we look back on our answer choices, the correct answer is answer choice. C thanks a lot for watching everyone. I hope this video helped.

Solve each triangle. See Examples 2 and 3.

a = 9.3 cm, b = 5.7 cm, c = 8.2 cm

Key Concepts

Triangle Classification and Properties

Law of Cosines

Triangle Solving Techniques

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