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# Law of Cosines Formula

Law of Cosines Formula is used to calculate the unknown sides and angles of a triangle. Given two sides and  angles (SAS) or three sides (SSS) of the triangle, the unknown side and angles can be calculated using the cosine law.

The Law of Cosine Formula is,

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 c2 = a2 + b2.

 Related Calculators Cosine Law Calculator Calculate Cosine Calculator with Sine Cosine and Tangent

## 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 = 9 cm and the angle A = 45o. Calculate the unknown sides and angles ?
Solution:

Given,
b = 7 cm
c = 9 cm
A = 45o

The law of cosines formula is,
a2 = b2 + c2 - 2bc cos A
a2 = (7 cm)2 + (9 cm)2 - 2(7 cm)(9 cm) cos 45
a2 = 49 cm2 + 81 cm2 - (126 cm2 $\times$ 0.707)
a2 = 49 cm2 + 81 cm2 - 89.082 cm2
a2 = 40.918 cm2
a = 6.397 cm

b2 = a2 + c2 - 2ac cos B
(7 cm)2 = (6.397 cm)2 + (9 cm)2 - 2(6.397 cm)(9 cm) cos B
49 cm2 = 40.918 cm2 + 81 cm2 - (115.146 cm2 $\times$ cos B)
115.146 cm2 $\times$ cos B = 40.918 cm2 + 81 cm2 - 49 cm2
115.146 cm2 $\times$ cos B = 72.918 cm2

cos B = $\frac{72.918\ cm^{2}}{115.146\ cm^{2}}$

cos B = 0.633
B = 50.71o

c2 = a2 + b2 - 2ab cos C
(9 cm)2 = (6.397 cm)2 + (7 cm)2 - 2(6.397 cm)(7 cm) cos B
81 cm2 = 40.918 cm2 + 49 cm2 - (89.558 cm2 $\times$ cos B)
89.558 cm2 $\times$ cos B = 40.918 cm2 + 49 cm2 - 81 cm2
89.558 cm2 $\times$ cos B = 8.918 cm2

cos B = $\frac{8.918\ cm^{2}}{89.558\ cm^{2}}$

cos B = 0.1
B = 84.29o

Question 2: A triangle ABC has the side length as BC = 15 cm; CA = 20 cm and the angle A = 30o. Calculate the unknown sides and angles ?
Solution:

Given,
BC = b = 15 cm
CA = c = 20 cm
A = 30o

The law of cosines formula is,
a2 = b2 + c2 - 2bc cos A
a2 = (15 cm)2 + (20 cm)2 - 2(15 cm)(20 cm) cos 30
a2 = 225 cm2 + 400 cm2 - (600 cm2 $\times$ 0.866)
a2 = 225 cm2 + 400 cm2 - 519.6 cm2
a2 = 105.4 cm2
a = 10.266 cm

b2 = a2 + c2 - 2ac cos B
(15 cm)2 = (10.266 cm)2 + (20 cm)2 - 2(10.266 cm)(20 cm) cos B
225 cm2 = 105.4 cm2 + 400 cm2 - (410.64 cm2 $\times$ cos B)
410.64 cm2 $\times$ cos B = 105.4 cm2 + 400 cm2 - 225 cm2
410.64 cm2 $\times$ cos B = 280.4 cm2

cos B = $\frac{280.4\ cm^{2}}{410.64\ cm^{2}}$

cos B = 0.68
B = 46.94o

c2 = a2 + b2 - 2ab cos C
(20 cm)2 = (10.266 cm)2 + (15 cm)2 - 2(10.266 cm)(15 cm) cos B
400 cm2 = 105.4 cm2 + 225 cm2 - (307.98 cm2 $\times$ cos B)
307.98 cm2 $\times$ cos B = 105.4 cm2 + 225 cm2 - 400 cm2
307.98 cm2 $\times$ cos B = -69.6 cm2

cos B = $\frac{69.6\ cm^{2}}{307.98\ cm^{2}}$

cos B = - 0.23
B = 103.06o

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