When given a problem like this, drawing a picture helps *A LOT*. Unfortunately, with only a text box, it's hard to show it.
But what we are given are distances between three points, A to B, B to C and A to C. When put together these make three sides (or SSS). To solve this triangle we need to use the Law of Cosines to find one angle then either the Law of Cosines or the Law of sines for as second angle. The third angle comes from subtracting the sum of the two from 180.
Let's try to find the angle between AB and BC - or angle B.
By the Law of Cosines,
cos B = AB² + BC² - AC² / 2(AB)(BC)
= 36.318² + 37.674² - 11.164² / 2(36.318)(37.674)
= 1318.997 + 1419.330 - 124.634 / 2736.489
= 2613.693 / 2736.489
= 0.95512
Through a calculator's inverse cosine (cos⁻¹) key, we have the angle as 17.229 degrees. Thus, angle B is 17.229 degrees.
To find another of the angles, we can use the Law of Sines or the Law of Cosines. The choice is yours, but let's use Law of Cosines a second time. We will use it to find angle A.
cos A = BA² + AC² - BC² / 2(BA)(AC) Notice by writing AB as BA, it tells us to make BA and AC the sides including angle A.
cos A = 36.318² + 11.164² - 37.674² / 2(36.318)(11.164)
cos A = 1318.997 + 124.634 - 1419.330 / 810.908
cos A = 24.301 / 810.908
cos A = 0.029968
Again, by a calculator's inverse cosine key, the measure of angle A is 88.28267 degrees.
Since the sum of the angles of any triangle is 180 degrees, we can find angle C by subtracting our two found angles from it.
angle A + angle B + angle C = 180
angle C = 180 - angle A - angle B by subtracting Angles A and B on both sides
angle C = 180 - 88.28267 - 17.229
angle C = 74.48833
We put these angles together - A being 88, B being 17, C being 74, and select choice A to solve the triangle.
