Solve each triangle ABC that exists.
A = 42.5°, a = 15.6...

Question

Solve each triangle ABC that exists.

A = 42.5°, a = 15.6 ft, b = 8.14 ft

Accepted Answer

Hello, everyone. We are asked to find the missing side lengths and angles of triangle ABC. We are given angle A is 49.12 degrees. Side A is 27.8 m. Side B is 21.75 m. Our answer choices are A angle B is 94.61 degrees. Angle C is 36.27 degrees. And side C is 36.65 m B. Angle B is 36.27 degrees. Angle C is 94.61 degrees and side C is 36.65 m. C angle B is 84.61 degrees. Angle C is 46.27 degrees and side C is 36.65 m D. Angle B is 104.61 degrees angle C is 26.27 degrees and side C is 36.65 m. All right. So I'm gonna use the law of science. We call the law of science is a ratio. So we have side A divided by the s of angle A is equal to the side B divided by the sine of angle B is equal to the side C divided by the sine of angle C. So I'm going to start with the portion that has A's and B's since that's what we're given in our problem. So I'm gonna start with 27.8 where the side A divided by the sine of angle A. So the sine of 49.12 degrees, that's gonna be equal to the side B. So 21.75 divided by the sign of angle B. So that's gonna be our unknown cross multiplying I have 27.8 multiplied by the sine of B and that will equal my other diagonal. So 21.75 multiplied by the sign of 49.12 degrees. I'm going to divide everything by 27.8 to get the S of B by itself. And I get that the sign of angle B, it is gonna be 21.75 multiplied by the sign of 49.12 degrees divided by 27.8. And I'm gonna place this in my calculator first, make sure your calculator is in degree mode. And when I do, I find that the sine of B equals zero 0.592. And then I'm gonna use the inverse sign. So the inverse sign of 0.592 will equal B. So placing that in my calculator, I get that B will equal 36.27 degrees. So that's part of my answer. I now know angle B from there, I'm going to find angle C and to find the angle C, I recall that a full triangle is 100 80 degrees. So I'm gonna subtract angle A and angle B from 180 degrees. So we'll have 180 degrees minus 49.12 degrees minus 36.27 degrees. And when we do that subtraction, we find that the angle at sea would be 94.61 degrees. So that's our c last thing we need to find is the side C. So the lower case C and I'm gonna go back to my law of signs for this and I'm gonna use a divided by the sign of angle A equals C divided by the sine of angle C. So plugging that in again, A was 27.8 divided by the sine of angle A. So the sine of 49.12 degrees, this will equal our unknown side C divided by the sine of the angle C we just found. So the sign of 49.61 degrees. And again, I'm going to cross multiply. So I'll have C multiplied by the s of 49.12 degrees equals 27.8 multiplied by the sine of 94.61 degrees to get sea by itself. I'm going to divide both sides by the sine of 49.12 degrees. So our side C will equal 27.8 multiplied by the sign of 94.61 degrees divided by the sine of 49.12 degrees. So I'm gonna place that in my calculator to find out what are like this. And I get that C will equal 36.65 and we were working with meters. So that's the length of side C, so looking at our answer choices, I want B to be 36.27 degrees. C to be 94.61 degrees and C to be 36.65 m. And that matches answer choice B have a nice day.

Solve each triangle ABC that exists.
A = 42.5°, a = 15.6 ft, b = 8.14 ft

Key Concepts

Law of Sines

Triangle Sum Theorem

Ambiguous Case of the Law of Sines

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