Let us draw the triangle
here the sides a= 37.674 miles
b= 11.164 miles
c= 36.318 miles
Lets use the cosine rule to solve for the angles
( we cannot use the sine law since we do not have the measure of any of the angles)
The cosine law
[tex] c^{2} = a^{2} +b^{2} -2ab cos C [/tex]
Let us plug in the values
[tex] (36.318)^{2} = (37.674)^{2} + (11.164)^{2} - 2(37.674)(11.164). Cos C [/tex]
[tex] 1318.997 = 1419.33 + 124.634 - 841.184. Cos C [/tex]
[tex] 1318.997 = 1543.964 - 841.184. Cos C [/tex]
[tex] 841.184. Cos C = 1543.964-1318.997 [/tex]
[tex] 841.184. Cos C = 224.967 [/tex]
[tex] Cos C = \frac{224.967}{841.184} [/tex]
[tex] cos C = 0.267 [/tex]
C = 74.48°
We can use the sine law to calculate the value of angle A
[tex] \frac{a}{SinA} = \frac{c}{Sin C} [/tex]
[tex] \frac{37.674}{Sin A} = \frac{36.318}{Sin 74.48} [/tex]
[tex] Sin A = \frac{37.674. sin 74.48}{36.318} [/tex]
[tex] Sin A = \frac{37.674 X 0.963}{36.318} [/tex]
[tex] Sin A = 0.998 [/tex]
A= [tex] sin^{-1} (0.998) [/tex]
A = 87.38°
Now we can easily find the third angle B by subtracting angle A and C from 180°
[tex] B = 180-(74.48 + 87.38) [/tex]
B = 180-161.86
B = 18.14°
Hence we have all the three angles ( attached figure)
