Solve each triangle. See Examples 2 and 3.

a ...

Question

Solve each triangle. See Examples 2 and 3.

a = 965 ft, b = 876 ft, c = 1240 ft

Accepted Answer

Welcome back. Everyone in this problem. We want to find the missing measurements of the triangle with side P being 512 m. Q 603 m and R 312 m. For our answer choices, A says angle P is 90 degrees six minutes. Angle Q 58 degrees 45 minutes and angle R 31 degrees nine minutes. B says angle P is 58 degrees six minutes. Q is 90 degrees 45 minutes and R is 31 degrees nine minutes. C says P is 31 degrees nine minutes. Q is 90 degrees 45 minutes and R is 58 degrees six minutes. And D says P is 58 degrees six minutes. Q is 31 degrees nine minutes and R is 90 degrees 45 minutes. Now, in order for us to find the missing measurements of the triangle, let's first try to draw the triangle. Let's get an idea of what's really going on here. And again, our triangle doesn't have to be perfect, but it just gives us an idea of where all of our measurements are. OK. So, so far Scarlets jangle, P QR, that's a long P. Let me bring, write that properly here. So that's P QR, OK. Then if this is, if this is angle P based on how we name triangles, the side opposite to an P will also be called P but a common P and we know that P equals 512 m. Likewise, if this is anger cue, then the opposite side. Q would be in front of it where Q is 603 m. OK? And that leaves us with side R being here to the left where side R is 312 m. OK? And now we're trying to find our unknown angles. PQ and R now let's think about it. We have three sides and we want to figure out the angles. What do we know that relates to three sides of a triangle? Well, if we think about it, if we can recall the cosine law, recall the cosine law tells us that if we have two sides for argument's sake, let's say sides P and Q and the anger between them an R, then we should be able to use that to solve for the side that's opposite to anger R. But that works both ways. In other words, we can solve for the angle if we know all three sides. So by the law of cosines, by the law, of course, science, it usually tells us that R squared is going to be equal to P squared plus Q squared minus two PQ multiplied by the cosine of R. But we're saying if we know RP and Q, we can transpose to eventually solve for R. So let's try to make R the subject of our equation here. That is an R and then see if we can solve for the value of R. OK. Now let's start by adding two P QR to two P QC R to both sides and subtracting R squared. And no, that gives us two PQ class I, OK, being equal to P squared plus Q squared minus R squared. OK. Now, if that's the case, that means the core sign of our is going to be equal to P squared plus Q squared minus R squared divided by two PQ. OK. And no, let me move a bit to the right here. If we want to solve for R, we're saying the cosine of some angle equals an expression which tells us then that the angle must be equal to the inverse cosine of that expression. In other words, R is equal to the inverse cosine of P squared plus Q squared minus R squared all divided by two PQ. No, we have the values for PQ and R so we can substitute to solve for the value of an R because no, this tells us that R is going to be equal to the inverse cosine of 512 squared. That's the value of P plus 603 squared. That's the value of Q minus R squared, which is 312. So that's 312 squared all divided by two, multiplied by P 512 multiplied by Q 603. So to find the value of R, we can put this expression into our calculators. Now, when you do that, what do you find? Well, you should find that R, OK is approximately equal to 31 degrees and nine minutes. OK? To the nearest minute. So R is 31 degrees and nine minutes. So we can put that value of R into our diagram or, or a sketch. I wouldn't say it's a diagram. Let me say it's a sketch. No, we know are, it's 31 degrees nine minutes. OK? And now that we have the value of R, let's see if we can use it to help us to find any of the other angles. Now, the value of R now shows us that we have a pair, a known pair of anger and the opposite side. If we want to figure out the value of angle P, notice that we know the side opposite to angle P, we know the value of side P. Now, what law can help us when we have a known pair of known pair of angle and size and we want to figure out an unknown angle. Well, we can think 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 this triangle. So if we can figure out what that ratio is, then we will be able to figure out our unknown side. OK. So let's do that here with side R and side P to figure out angle P. So by the law of science, we know that the sign of anger P, the side P is going to be equal to the sine of anger, R to side R. We don't know P but we know that side P is 512 meters. So the sine P to 512 or sine P divided by 512 equals the sine of R which we just found to be 31 degrees and nine minutes divided by uh the value of side R which is 312 m. Therefore, that means the sign of P, it is going to be equal to 512 multiplied by the sine of 31 degrees nine minutes divided by 312. OK. And again, if we have the sine of an angle equal to some expression, then we can figure out what that anger is by using the inverse sine. In other words, P is going to be equal to the inverse sine of that expression, which is 512 sine 31 degrees nine minutes divided by 312. So that's the value of P. So let's figure that out by putting it into our calculators. Now, when we do that, we should get the value of P to be approximately equal to 58 degrees and six minutes to the nearest minute. So now we have a value for P as well. Hm. So we figured out P OK, we figured out R let's put that in our diagram here. We now know that he is 58 degrees and six minutes. So now all we have left is to figure out the value of angle Q. Now how can we figure that out? Well, think about it here. OK. Here we have our two angles and the third angle is unknown. What do we know that relates the angles within a triangle? We'll recall that by the angle sum property angles in a triangle sum to 180 degrees. So if we can subtract these two angles from 180 degrees, we should be able to solve for Q. So let's do that. So solving for Q, let me make some space again on the right here. OK. Solving for QE using the anger some property of triangles. OK? And no, that means Q is going to be equal to 180 degrees minus P and R, in other words, minus 58 degrees and six minutes minus 31 degrees and nine minutes. OK. So that would be the value of Q. Now, when we put that into a calculator. When we work that out, we should find that cu equals 90 degrees and 45 minutes. Therefore, angle P is 58 degrees six minutes. Q is 90 degrees 45 minutes and R is 31 degrees nine minutes. If we look back on our answer, choices, the correct answer would be choice. B thanks a lot for watching everyone. I hope this video helps.

Solve each triangle. See Examples 2 and 3.

a = 965 ft, b = 876 ft, c = 1240 ft

Key Concepts

Law of Cosines

Triangle Angle Sum Property

Triangle Classification by Sides

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