The **Law of Cosine Formula** is,

$${b}^{2}={a}^{2}+{c}^{2}-2(ac)Cos\phantom{\rule{thickmathspace}{0ex}}B$$

$${c}^{2}={a}^{2}+{b}^{2}-2(ab)Cos\phantom{\rule{thickmathspace}{0ex}}C$$

The cosine law can be derived out of Pythagoras Theorem.

The Pythagorean theorem can be derived from the cosine law. In the case of a right triangle the angle, θ = 90°. So, the value of cos θ becomes 0 and thus the law of cosines reduces to

${c}^{2}={a}^{2}+{b}^{2}$

## Law of Cosines Problem

Some solved problem on the law of cosines are given below:

### Solved Examples

**Question 1:**Given the sides of the triangle b = 7 cm; c = 8 cm and the angle A = 45

^{o}. Calculate the unknown sides and angles ?

**Solution:**

b = 7 cm

c = 8 cm

A = 45

^{o}

^{2}= b

^{2}+ c

^{2 }– 2bc cos A

^{ }a

^{2}= (7 cm)

^{2}+ (8 cm)

^{2 }– 2(7 cm)(8 cm) cos 45

^{ }a

^{2}= 49 cm

^{2}+ 64 cm

^{2 }– (112 cm

^{2 }x 0.707)

a

^{2}= 49 cm

^{2}+ 64 cm

^{2 }– 79.18 cm

^{2}

a

^{2}= 33.82

b^{2}= a^{2} + c^{2} – 2ac cos B72 = (5.82)^{2} + 8^{2} – 2(5.82)(8) cos B

49 = 33.8724 + 64 – 93.12 cos B

93.12 cos B = 48.8724

Cos B = 48.8724/93.12

B = 58.3o

c^{2} = a^{2}+ b^{2} – 2ab cos C

8^{2} = (5.82)^{2} + 7^{2} -2(5.82)(7) cos C

64 = 33.8724 + 49 – 81.48 cos C

81.48 cos C = 18.8724

Cos C = 18.8724/81.48

C = 76.6^{o}