Solve each triangle. See Examples 2 and 3.

C ...

Question

Solve each triangle. See Examples 2 and 3.

C = 45.6°, b = 8.94 m, a = 7.23 m

Accepted Answer

Welcome back. Everyone in this problem, we want to find the missing measurements of the triangle where R is 23.7 degrees. Side Q is 4.72 inches long and side P is 2.75 inches long for our answer choices, A says angle P is 26.7 degrees. Anger Q is 129.6 degrees and side R is two point or six inches. B says P is 26.7 degrees. Q is 129.6 degrees and R is 4.26 inches res respectively. C says angle P is 100 and 26.7 degrees. Q is 29.6 degrees and the side R is 2.46 inches and D says P is 129.6 degrees. Q is 26.7 degrees and side R is 2.46 inches. Now, for us to figure out what this looks like, it would be good for us to try and sketch what's really going on here. So let's say that we have our triangle. OK? And our triangle is triangle P QR. So uh I think I'm gonna put P here. So let's say this is PQ. Uh Well, you know, I mean, and we don't have to get the sketch right the first time because we just want to have an idea where to place all this information. OK. So we have P QR and in the same way that we usually name triangles. If this is angle P, then the side opposite to it is also referred to as P as a common P. Likewise opposite to angle Q will be side Q and opposite to angle R is going to be side R. OK. And we already know we already know that angle R is 23.7 degrees. So let's put that in there. Let me make it a bit, let me make it a bit more clear. That's 23.7 degrees. We know that side Q is 4.72 inches. OK? And side P is 2.75 inches. OK? I think I'm gonna bring that over a bit just to show that it's right here. So we have this information already. And now we're trying to find an P anger Q and side R. So first, let's think about the information we have notice there that we have two sides, the side Q and side P and the angle between them angle R. Now when we have two sides and an angle between them, what law can we use where we can use the law of cosines and the law of cosines or we can use that law to help us to find a side that's opposite to our angle. In this case, that would have been side R. So let's use the law of cosines to figure out the value of side R. So by the law of cosines, by the law, of course, science, we know then that R squared is going to be equal to P squared plus Q squared. Mean it's a little bit like the Pythagorean theorem with a little extra minus two multiplied by PQ multiplied by the cosine of R. No substituting our values here. That means R squared is going to be equal to 2.75 inches squared plus Q squared, 4.72 inches squared minus two, multiplied by 2.75 multiplied by 4.72 multiplied by the cosine of the angle between them, which is 23.7 degrees. OK. So that's how we could use our law of cosines. A sulfur R. Now let's figure out the value of our expression here using a calculator. Now, when we put that expression into our calculator, we should find that R squared is equal to 6.0702 square unit. Therefore, that means R OK, R is going to be equal to the square root of that value 6.0702 which would be equal to 2.46 inches to two decimal places. Because notice here, all of our answers for the length of our side in the list of answers was written to two decimal places. So I just put it to two decimal places. So that would be the value of R. So now if we go back to our triangle, we can put the value of R to be 2.46 inches. OK? So we know what the value of R is. What else do we need to figure out? Well, here, let me clear what we were trying to figure out before. So now we know what that is. We now have three sides and we have two unknown angles. So we can try to figure out any of the unknown, unknown angles because importantly, we have a pair of anger and side and when we have a pair of known angle and side, and we want to find an unknown angle where we know the side that's opposite to the angle, then we can use the law of science because the law of science tells us that the s of an angle to its opposite side is the same ratio throughout the triangle. So let's say that we were trying to find uh angle angle P for, for argument's sake right? Then to find an P, OK? To find anger P, given that we know the ratio, well, we know angle R and side R we can use that to help us figure out an P because we know side P. OK? So let's use the law, of course, the law of science because no, by the law of science, I the law are signed then the ratio of the sine of P to side P is going to be the same as the sine of R to side R. OK? So no, that means the sine of P. Let's put our values here divided by the length of P which is 2.75 inches is going to be equal to the sine of R which is 23.7 degrees divided by the length of R which we just found to be 2.46 inches. Therefore, that means the sine of P is going to be equal to 2.75 multiplied by the sine of 23.7 degrees divided by 2.46 which means then that P is going to be equal to the inverse sine of two point of that expression 2.5 S 23.7 degrees divided by 2.46. OK. No, to solve for P, we can put that into our calculator, we can put that value into our calculator here. OK? And when we do that, you should get angle P to be approximately equal to 26.7 degrees to the nearest 10th or to the nearest uh decimal one decimal place. So we've rounded it to one decimal place like the rest of the answers uh in our list of answers are, so we found a value for angle P. So let's put that value in our diagram here. So now we know that P OK, P is equal to 26.7 degrees. So now that we have angle P, we have side are the only thing left to find is angle Q. So now how can we figure out the value of Anger Q? Well, there are a lot of different ways. I mean, we could use the law of signs, we could use the law of cosines. But let's make it a little easier for ourselves. Notice that we have two angles and we want to find the third angle in a triangle. What do we know about angles in a triangle? We recall that the sum of angles in a triangle equals 180 degrees. So we could use that idea to figure out angle Q. So let's do that. So using what we know about triangles, OK. Since angles in a triangle some to 180 degrees, then Anger Que is going to be equal to 180 degrees minus the sum of the two other angles in the triangle that is 26.7 degrees angle P and 23.7 degrees. That would have been an R. Now, therefore, when we, when we subtract both of those angles from 180 degrees. We get anger Q to be equal to 129.6 degrees. Therefore, angle R, sorry, not angle R. We knew that already. Angle P is 26.7 degrees. Angle Q is 129.6 degrees and the side R is 2.46 inches. If we go back to our list of answers, it is the correct answer. Thanks for watching everyone. I hope this video helped.

Solve each triangle. See Examples 2 and 3.

C = 45.6°, b = 8.94 m, a = 7.23 m

Key Concepts

Law of Sines

Triangle Properties

Trigonometric Functions

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