Find the unknown angles in triangle ABC for each triangle that...

Question

Find the unknown angles in triangle ABC for each triangle that exists.

A = 29.7°, b = 41.5 ft, a = 27.2 ft

Accepted Answer

Hello, everyone. We are asked to determine the measure of the remaining angles of triangle ABC. With the following measurements, we have that angle A is 22.9 degrees. Side B is 49.2 ft. Side A is 23.7 ft. Our answer choices are A B equals 53.9 degrees and angle C equals 103.2 degrees. Also we'd have the triangle where angle B equals 126.1 degrees. And angle C equals 31.0 degrees. B. Angle B would equal 53.9 degrees and angle C would equal 103.2 degrees. C angle B would equal 126.1 degrees and angle C would equal 31.0 degrees. D. Angle B would equal 43.9 degrees and angle C would equal 113.2 degrees. Also, angle B could equal 116.1 degrees and angle C could equal 41.0 degrees. Looking at this, I want to use the law of sines and I recall that the law of sines is a ratio where I have the side A divided by the sine of angle A equal to the side B divided by the sine of angle B equal to the side C divided by the sine of angle C. Here we are given A's and B's. So that's the section I'm going to start out working with. So the side A is 23.7 ft divided by the sine of angle A. So the sine of 22.9 degrees that will equal the side of B. So 49.2 divided by the sine of angle B, which is our unknown. So we're gonna find the measure of angle B. Here, I'm gonna start by cross multiplying. So we'll have 49.2 multiplied by the sine of 22.9 degrees. And that will equal our other diagonal which is 23.7 multiplied by the sine of B. I want to get the sine of B by itself. So I'm going to divide both sides by 23.7. So then I have that the sine of angle B equals 49.2 multiplied by the sine of 22.9 degrees divided by 23.7. At this step, I'm gonna put this into my calculator and I first want to make sure my calculator is in degree mode and then we'll see what we have from there. So using my calculator, I get the sign of angle B is 0.80 78. That's not telling me what the angle measures yet. So I'm going to now use the inverse sign of 0.8078. So I can find the angle B and with that, I get 53.9 degrees with the angle at B, it is 53.9 degrees. But since sin was positive, we know this could be in quadrant one or quadrant two. So this is B when we're in quadrant one, defined B, when we're in quadrant two, I need to do 180 degrees minus 53.9 degrees. And when I do that, I get 126.1 degrees. And just to check to see if this can be another triangle, I'm in due 180 degrees minus our value of a 22.9 degrees minus 126.1 degrees. And if this comes out to a positive value, we've found the measure of angle C and if it doesn't, we do not have a second triangle. So when I subtract this, I get an even 31.0 degrees. So we do have two triangles. And this is actually the measure of angle C for our quadrant two second triangle. So we know that we have two triangles in one of them. B is 53.9 degrees and the other B is 126.1 degrees and C is 31.0 degrees. So let's find out what quadrant one angle C would be. So I'm gonna use again that the interior angles of a triangle have a sum of 180 degrees and do 180 degrees minus angle A which is 22.9 degrees minus quadrant ones measured for angle B which is 53.9 degrees. And when I subtract that I get 103.2 degrees, and that's going to be our measure of angle C. So again, we have two triangles. The first one in quadrant one has a bee value of 53.9 degrees and a sea value of 103.2 degrees in quadrant two B is 126.1 degrees and C is 31.0 degrees. So those are our values and they appear to match answer choice. A have a good day.

Find the unknown angles in triangle ABC for each triangle that exists.

A = 29.7°, b = 41.5 ft, a = 27.2 ft

Key Concepts

Law of Sines

Ambiguous Case of the Law of Sines (SSA Condition)

Triangle Angle Sum Property

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