If the two sides and angles of the triangle are given, then the unknown side and angles can be calculated using the cosine law. Law of cosine is another formula used to find out the unknown side of the triangle.

The **Law of Cosine Formula** is,

\[\large a^2=b^2+c^2-2(bc)Cos\;A\] \[\large b^2=a^2+c^2-2(ac)Cos\;B\] \[\large c^2=a^2+b^2-2(ab)Cos\;C\]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:**

Given,

b = 7 cm

c = 8 cm

A = 45

b = 7 cm

c = 8 cm

A = 45

^{o}The law of cosines formula is: a

a

a

^{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}= 33.82a = 5.82 cm

b

^{2}= a^{2}+ c^{2}– 2ac cos B72 = (5.82)^{2}+ 8^{2}– 2(5.82)(8) cos B49 = 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}