Solve each triangle. See Examples 2 and 3.

A ...

Question

Solve each triangle. See Examples 2 and 3.

A = 41.4°, b = 2.78 yd, c = 3.92 yd

Accepted Answer

Welcome back. Everyone. In this problem, we want to find the missing measurements of the triangle with the measurements given where angle P of the triangle is 33.4 degrees. Side Q is 5.78 centimeters long and the side R is 4.21 centimeters long. For our answer choices. A says side P is 2.43 centimeters long. Angle, Q is 47 45.7 degrees, sorry. And angle R is 100.9 degrees. B says there are 2.43 centimeters, 100.9 degrees and 45.7 degrees respectively. C says there are 3.24 centimeters, 45.7 degrees and 100.9 degrees respectively. And D says they are 3.24 centimeters, 100.9 degrees and 45.7 degrees respectively. No. In order to find the missing measurements of the triangle, let's try to draw it to get a sense of what's going on here. So we have a triangle, P QR, OK? And four or a triangle four or a triangle. OK. Then we have our side P you know what I feel like I could have drawn that a bit bigger. Let me try it a little bit bigger here. OK. And for our triangle, we know that angle P, it's 33.4 degrees. OK? Uh We have uh Q and we have an R which we don't know. OK. These are the ones that we don't know. And we already know that side Q, which is opposite to anger. Q is 5.78 centimeters. So I should write, I should even say that Q equals just to show that we're talking about Q. So Q equals 5.78 centimeters and side R is equal to 4.21 centimeters. OK. And we are trying to find the missing measurements that is we're trying to find side P angle Q and angle R. So let me put those in red here. These are the ones that we don't know. Now, first, let's start with what we do know. OK. So we have an anger and we know two sides. We know the two sides that make the anger. OK. So are there any laws right away that we could use to help us? Well, here, if we have two angles, OK, if we have two angles are two sides rather and the anger between them, then we can use the law of cosines to find the length of the side opposite to our angle. In this case, we can use the law of cosines to find the length of side P OK. So solving for P so for a side P using the law of cosines, then by the law of cosines, be squared, be squared is going to be equal to Q squared plus R squared. Remember it's just like the Pythagorean theorem but with just a little extra on it, that is minus two QR multiplied by the cosine of anger P. So in this case, then that means P squared is going to be equal to 5.78 centimeters squared plus 4.21 centimeters squared minus two, multiplied by 5.78 multiplied by 4.21 multiplied by the cosine of the angle between them, which is 33.4 degrees. OK. Now, to solve for P squared, we can put that into our calculator. And when we put that into our calculator, OK? Notice that all of our answers were rounded to two decimal places. That is our answers for P, then we should get P squared to be equal to, well, to be safe, let me put it to a little extra. So we should get it to be 10.50 to 5. OK. So that should be our value for P squared. Therefore, that means P is going to be equal to the square root of that result. And again, when we put this into our calculator, we should find that P is going to be approximately equal to 3.24 centimeters, rounded to two decimal places. OK. So that's the length of side P. So P is 3.24 centimeters. Know that we have the value of P, what else can we find? Well, no, we have three sides. Let me get rid of the uh drawings that I had here because no, we have three sides. OK. And if we have three sides, what could we use to figure out any of the angles? What are you thinking? Well, here we could solve for angle R, we could solve for Q two. But let's choose angle R. Now here for angle R notice that so far we have a pair of sides, OK? We have a pair of angle and side and we have an unknown angle but we know the side that's opposite to it. So in this case, we could use the s the law of science because 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 if this ratio is the same, we can use it to solve for R. OK. So let me put that in red. OK? No, by the law of science, by the law of science, OK? Then this tells us that the ratio of sr to R OK ra off sign R to R is going to be the same as the ratio. Of the sin of which side did we choose P? So it's gonna be the same as the ratio of the sine of angle P to side P. Now we can solve because that means the sine of R divided by the length of R which is uh 4.21. OK. It is going to be equal to the sine of angle P which is 33.4 degrees divided by the length of P which we are found to be 3.24 centimeters. Therefore, that means the sign of R, if we multiply both sides by 4.21 the sine of R is going to be 4.21 multiplied by the sine of 33.4 degrees divided by 3.24 which therefore means that R, OK, because if we want to figure out R, the sine of an angle equals this expression. So to figure out the angle, we can find the sine inverse of that expression. So R is going to be the inverse sine of 4.21 multiplied by the sine of 33.4 degrees divided by 3.24. So to figure out the value of R, we can put that into our calculator. OK? Now when you do that, what do you find? Well, you should find that R again, if we round it to a decimal place based on the other angles that we have, you should find that R equals 45.7 degrees. So that would be the value of anger R. OK. So, so far we've found the valley oxide P 3.24 centimeters. We just found the value of angle R. Let's again, let me clear these here. And uh let's put R inside our angle here. We now know that R is 45.7 degrees. The last thing that we have left to find is the value of angle Q. Now again, what can we use to help us? Well, I mean, we have a lot of information. No, we could use the law of signs, the law of cosines, but we already have two angles inside our triangle and recall that the sum of angles in a triangle is 180 degrees. So if we have these two angles, we can use the anger some property to find the third angle. So let's go ahead and do that. OK? So now let me put a line here solving for, say what, what are we solving for? Again, we're solving for Q. So solving for Q, OK, buy the Un Song property of triangles. We know that Q plus the value of our other two angles, 33.4 degrees and 45.7 degrees is going to be equal to 180 degrees. Therefore, Q is going to be equal to 180 degrees minus the sum of 33.4 degrees and 45.7 degrees. OK. So we can subtract both of those from 180. And when we do that, we should find that it's equal or the value of Q is equal to 100.9 degrees. Therefore, side P, side P equals 3.24 centimeters. Angle Q is 100.9 degrees and angle R is 45.7 degrees. If we look back on our answer choices, the correct choice is answer choice. D thanks a lot for watching everyone. I hope this video helped.

Solve each triangle. See Examples 2 and 3.

A = 41.4°, b = 2.78 yd, c = 3.92 yd

Key Concepts

Law of Sines

Triangle Sum Theorem

Side-Angle Relationships

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