Solve each triangle ABC that exists.
A = 96.80°, b = 3.5...

Question

Solve each triangle ABC that exists.

A = 96.80°, b = 3.589 ft, a = 5.818 ft

Accepted Answer

Hello, everyone. We are asked to find the missing side lengths and angles of triangle ABC we are told angle A is 107.52 degrees. Side B is 5.327 ft. Side A is 8.965 ft. Our answer choices are A angle B is 34.52 degrees. Angle C is 37.96 degrees and side C is 12.48 ft B. Angle B is 37.96 degrees. Angle C is 34.52 degrees and side C is 5.78 ft C angle B is 34.52 degrees. Angle C is 37.96 degrees and side C is 5.78 ft D, no triangle exists. All right. So we are gonna use the law of signs and recall that the law of signs basically says that we have the side divided by the sine of the angle. So we have a side A divided by the sine of A but equal side B divided by the sine of angle B. What do you call the uh side C divided by the sine of angle C. So with what we're given so far, we're given two things about A, we know the side A and the angle A and we know the length of side B. So I'm going to start out using the section where I have A divided by the sine of A equals B divided by the sine of B. So I'll have A which is 8.965 divided by the sine of angle A which is 107.52 degrees. And that's going to equal side B. So 5.327 divided by the sign of side B which is our unknown at the moment. So let's find out what we can. I'm gonna find out what is the sign of 107.52 degrees. And I'm gonna plug that in. Nothing else is changing. So I have 8.965. It's gonna be divided by, and I'm rounding this to four places, 0.9536 and that equals 5.327 divided by the sine of B. So now I cross multiply so cross multiplying my numbers. So 0.9536 multiplied by 5.327 will equal, I'm gonna round a little 5.0798 and that's gonna be equal to cross multiplying in the opposite direction which will be 8.965 multiplied by the S of B, get the sign of B by itself. I'm gonna divide both sides by 8.965. I'm gonna bring that right over here. And when we do, we find that the sine of B will equal. And again, I'm rounding a little bit, zero point 5666. All right. Well, I want to find out what B is. So now I'm going to use the inverse sign. So I'm going to find the inverse sign of 0.5666. And that will tell me the value of angle B, make sure when you're doing this, your calculator is in degree mode. So I find that B will equal 34.52 degrees. And I did round to two decimal places because that's what all of our answer choices had us rounding to. So now I know two of my angles. So let's find the third angle. So if I want to find angle C, I can use the fact that a full triangle is 100 80 degrees and I will subtract the value of angle A. So 107.5 th I'm sorry, 52 degrees, then I will also subtract what we just found for angle B. So 34.52 degrees. And when I do, I get that angle C would have a measure of 37.96 degrees. So we know C and B now we need to just find out what is the length of side C Well, now that we know the angle for sea and the angle for B we can use our law of signs again. So I'm going to stick with the information for A. So I'm going to use a divided by the sine of A and I'm going to say that this will equal C divided by the sine of angle C. So the length of side A was 8.965 and that was divided by the sine of 107.52 degrees. And that's gonna equal our unknown, which is side C and it'll be divided by the sign of Anglesey. We just found so 37.96 degrees. So I've already figured out what the sign of 107.52 degrees is earlier in this problem. So I'm gonna fill that in. So I have 8.965 divided by 0.9536 equals C divided by, I need to find the sign of 37.96 degrees. And that is 0.6151. And I rounded a little again, we're going to cross multiply. So I'll have 0.9536 multiplied by C and then 8.965 multiplied by 0.6151 which equals 5.5145 dividing both sides by 0.9536 to get C by itself. We find that side Sea is 5.78 and I'm stopping at two decimal places. Rounding to that because that's what our answer choices look like. Feet. So side C is 5.78 ft. So looking at all my answers, this looks like it is answer choice. C angle B is 34.52 degrees. Angle C is 37.96 degrees and side C is 5.78 ft. Have a nice day.

Solve each triangle ABC that exists.
A = 96.80°, b = 3.589 ft, a = 5.818 ft

Key Concepts

Law of Sines

Triangle Properties

Angle Measurement

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