Application of Cosine Rule

Example

Solve the triangle in which a = 4.5m, b = 5.3m, and C = 112°

Solution:

To find length c we will use the cosine rule formula.

c² = a² + b² – 2ab Cos C

c² = 4.5² + 5.3² – 2 × 4.5 × 5.3 × Cos 112°

c²= 20.25 + 28.09 + (47.7 × 0.3746)

c²= 48.34 + 17.87

c² = 66.27

c = \( \scriptsize \sqrt {66.21} \) = 8.1m

Cos A = \( \frac{b^2 \: + \: c^2 \: – \: a^2}{2bc}\)

= \( \frac{5.3^2 \: + \: 8.1^2 \: – \: 4.5^2}{2 \: \times \: 5.3 \: \times \: 8.1}\)

= \( \frac{28.09 \: + \: 65.61 \: – \: 20.25 }{85.86}\)

= \( \frac{73.45}{85.86}\)

Cos A = 0.8555

A = Cos^-1(0.8555)

A = 31.18°

Go ahead and find B