In Exercises 9–16, solve each triangle. Round lengths to the near...

Question

In Exercises 9–16, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree.

B = 80°, C = 10°, a = 8

Accepted Answer

Hey, everyone in this problem, we're asked to find the missing side lengths and the angle of the triangle ABC. And to express the lengths to one decimal place and the angle to the nearest degree, we're told that the angle B is equal to 70 degrees. The angle C is equal to 20 degrees and side length A is equal to 11. We have four answer choices A through B each containing a different value for the three things. We need to find side B side C and angle bay. Now let's get started. And the first thing we're gonna do is deal with the angles. OK? We know that the angles in a triangle have to add up to 180 degrees. We were given two angles in the problem. So that's gonna allow us to find that third angle right off the bat. So again, recall the angle A angle, what's Anglesey is equal to 180 degrees? OK. So we have that angle A plus angle B which we're told is degrees plus angle C which we're told is 20 degrees is equal to 180 degrees. OK? On the left hand side, if we simplify 70 plus 20 degrees is equal to 90. We also track that from both sides to get our angle A which is equal to 90 degrees. OK. So we found that our angle A is equal to 90 degrees. So that's great. If we look at our answer choices, option A has angle A being 80 degrees. Option B has angle A being 95 degrees and neither of those are correct. And so we can actually eliminate both of those options are left with AC or D. Yeah, we know that angle A is 90 degrees. So what we have here is a right triangle. Let me draw it out. So we have a right triangle here. We don't know all the details, but we know some. Yes, we know we have a right angle and this is angle A across from angle A is gonna be side A which we're told in the problem that the length of 11, then we have angle B we're told is 70 degrees across from angle B is gonna be side B. Then we have angle C and sight see where angle C is 20 degrees. So we have this triangle and now what's left to find is side B and side C. And there's a couple of ways we can do this. OK? Because we have a right triangle. We have two options. The first is to use our law of science. OK? And look at the ratio of side length and corresponding angle or we can just use some right angle triangle math. Since we know that this is a right triangle and we're gonna take that second approach, it's gonna be just a little bit simpler in this case. So let's think about C first. OK. If we want to find side length C, we can relate that to angle C through the sign. OK. So if we say that sign of angle C, that's gonna be the opposite side, which is side, see divided by the hypotenuse, which is side A that tells us that C is gonna be equal to sine of angle C 20 degrees multiplied by A which is 11. And if we work this out on our calculator, we get that side C, there's gonna be 3.76 units. Now we can do the same for B OK. Stein of angle B is going to be equal to side B divided by side A. OK. The opposite side divided by the hypotenuse. So we get that B is equal to sign of degrees multiplied by 11, which gives us 10 0.3366 units, approximately four side B. And we can, could have used other trig functions as well. We could have used the cosine, we could have used the tangent because we knew one side and we knew actually we couldn't use the tangent because we didn't have these sides. But we knew the hypotenuse. So we could have used sine or we could have used cosine. Either one would have been OK in this case. All right. So we found our two side links and we went around to one decimal place. So we found that sideline C is gonna be about 3.8 units and side length B is gonna be about 10.3 units which corresponds with answer choice. C thanks everyone for watching. I hope this video helped see you in the next one.

In Exercises 9–16, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. B = 80°, C = 10°, a = 8

Key Concepts

Law of Sines

Triangle Sum Theorem

Rounding in Trigonometry

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