Law of Cosines, Case 1

When to use ‘Law of Cosines’ for triangle

QUESTION: A triangular park has side lengths that are 200 m, 155 m, and 172 m in length. What is the size of the angle between the two longest sides?

SOLUTION:

Let a = 200 m

b = 155 m

and c = 172 m

B is the angle between
the longest sides a and c

We can find angle B by

Law of Cosines

b² = a² + c² – 2 a c Cos B

Put values

(155)² = (200)² + (172)² – 2(200)(172) Cos B

24025 = 40000 +29584 – 68800 Cos B

24025 = 69584 – 68800 Cos B

68800 Cos B = 69584 – 24025

68800 Cos B = 45559

Cos B = $\frac{45559}{68800}$

B = $Cos^{-1}\frac{45559}{68800}$

B = $48.53^0$

Thus the size of the angle between the two longest sides
200 m and 172 m is 48.53 degrees.