Find the measures of angles A, B, and C if: a = 31, b= 58, and c = 36.

Answer 1

Since this is a non-right triangle with all 3 side measurements given, you will need to use the Law of Cosines at the start to find one of the angles. Once you have one angle measure, you can then use Law of Sines to find a second angle measure, and then for the 3rd angle, you can simply use the Angle Sum Theorem for triangles.

Let's try for angle A to start. By Law of Cosines, we have the equation 31² = 58² + 36² - 2(58)(36) cos A. Doing the arithmetic and isolating the "cos A" part of the equation, we have cos A = .886. Using the cosine inverse function on the calculator, this gives angle A as approximately 27.6 degrees.

Now using angle A with side A and then pursuing angle B, we can set up the Law of Sines as:

31/sin 27.6 = 58/sin B. By cross multiplication and then dividing to isolate sin B, we have sin B = .8668. Again using the inverse function but now for sine, we have angle B approx equal to 60 degrees. Note: there are two angle measures whose sine value is .8668. Since a measure of 60 degrees for angle B would force the 3rd angle to be the largest angle in the triangle ( 180 - (27.6 + 60)), this cannot be the case because the largest angle must be across from the longest side. Side b IS the longest side; therefore, angle B must be the biggest angle. So we need to go with the other angle measure whose sine value is also .8668. This would be a 120 degree angle.

Now by the angle sum theorem, 180 - ( 27.6 + 120) = 32.4 degrees, the measure of angle C.