Solve each triangle ABC that exists.
A = 38° 40', a = 9....

Question

Solve each triangle ABC that exists.

A = 38° 40', a = 9.72 m, b = 11.8 m

Accepted Answer

Hello, everyone. We are asked to find the missing side lengths and angles of the triangle. ABC. We are given the measure of angle A is 45 degrees 38 minutes. Side A measures 14.58 m. Side B measures 13.47 m. Our answer choices are A, the measure of angle B equals 41 degrees 20 minutes. Angle C measures 93 degrees one minute and side C measures 20.37 m B. Angle B measures 41 degrees 20 minutes. Angle C measures 93 degrees one minute and side C equals 25.15 m C. Angle B equals 93 degrees one minute. Angle C is 41 degrees 20 minutes and side C is 20.37 m D no triangle exists. So recall our law of signs which is a ratio from the side of A divided by the sine of angle A which is equal to the ratio of the side B divided by the sine of angle B which is equal to the side C divided by the sine of Anglesey. So looking at what we're given, I'm gonna start with my A's and B's and I'm gonna be looking for the angle B. So substituting in what we are told, the side A is 14.58 divided by the sine of angle A. So the sign of 45 degrees 38 minutes, that's gonna equal the side B 13.47 divided by the sign of angle B which is our unknown. I'm gonna cross multiply. So I get 14.58 multiplied by the sine of angle B is equal to my other diagonal, which will be 13.47 multiplied by the sign of 45 degrees. 38 minutes. I want to get the sine of B by itself. So I'm gonna divide both sides by 14.58. So I find that the sign of B equal 13.47 multiplied by the sine of 45 degrees 38 minutes divided by 14.58. So this is what I'm gonna put in my calculator first. Make sure your calculator is in degree mode. And also as a side note, when I wanna put the sign of 45 degrees 38 minutes into my calculator, I'm gonna enter it as the sign of open parentheses. 45 plus 38 divided by 60. That way, I don't have to round at any point. So when I do this, I get that the sine of B equals 0.6605 which is rounded slightly. Then I'm gonna use the inverse sign and do the inverse sign of 0.6605 to find the measure of B. And when I do that, I find that B equals 41.335 degrees. Now, unfortunately, for us, we can't stop there because our answer isn't in decimal degrees. It is in degrees and minutes. So I'm gonna break this down. So the angle B will be 41 degrees plus 0.335 multiplied by 60 minutes. So this works out for angle B to be 41 degrees, 20 minutes. So we know one of our angles. Now, we know A and B. So let's find the angle at sea. Recall that the sum of the interior angles of a triangle is 100 80 degrees. So I can say this is 100 and 80 degrees minus the measure of angle one minus the measure of angle. I'm sorry, not angle one angle A minus the measure of angle B and that will equal the measure of angle C. So 180 degrees minus 41 degrees, 20 minutes minus 45 degrees, 38 minutes. And this will equal 93 degrees one minute. So our measure of angle C is 93 degrees, one minute. All right, almost there. We still need to find the side. C I'm again gonna use the law of sign. I'm gonna take the portion that is a divided by the S of A equals C divided by the sine of C. So substituting in what I know 14.58 divided by the sign of 45 degrees. 38 minutes will equal my unknown side. Sea divided by the sign of 93 degrees one minute. Again, I'm gonna cross multiply. So I will have C is multiplied by the sine of 45 degrees 38 minutes and that'll equal my other diagonal which is 14.58 multiplied by the sign of 93 degrees one minute. I do want sea by itself. So I'm gonna divide both sides by 45. I'm sorry, the sign of 45 degrees 38 minutes. So C will equal 14.58 multiplied by the sine of 93 degrees one minute divided by the sign of 45 degrees 38 minutes. And I'm going to place this in my calculator. And again, I'm gonna use that little trick of dividing my minutes by 60 when I need to enter it. And when I put that all in my calculator, I find C to equal 20.37 and we're working with meters. So side C is 20.37 m. So checking my answer choices where B is 41 degrees, 20 minutes, C is 93 degrees one minute and side C is 20.37 m. I find that my answer is answer choice. A have a nice day.

Solve each triangle ABC that exists.
A = 38° 40', a = 9.72 m, b = 11.8 m

Key Concepts

Law of Sines

Angle Conversion

Triangle Properties

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