Law of Cosines is an equation relating the lengths of the sides of a cosine of one of its angles.

More About Law of Cosines

For any triangle ABC, where a, b, and c are the lengths of the sides opposite to the angles A, B, and C respectively, the Law of cosines states that:

a^{2} = b^{2}+ c^{2}- 2bc cos A
b^{2} = a^{2} + c^{2} - 2ac cos B
c^{2} = a^{2} + b^{2} - 2ab cos C
Law of cosines is also called as cosine rule or cosine formula.

Video Examples: Law of Cosines

Example of Law of Cosines

The figure below shows two of the six Law of Cosiness of a cube In triangle ABC, if a = 19, b = 12, and c = 10 are the lengths of the sides opposite to the angles A, B, and C respectively, then, by using law of cosines, the measure of angle A can be obtained this way:
a^{2} = b^{2}+ c^{2}- 2bc cos A
Cos A = b^{2}+ c^{2} - a^{2}/ 2bc
Cos A = 12^{2} + 10^{2} - 19^{2}/ 2(12) (10)
Cos A = - 0.4875
∠ A = 119°

Solved Example on Law of Cosines

Ques: In â–³DEF, if angle D = 46°, f = 10, and e = 17, then find the length of d to two significant digits

Choices:

A. 27
B. 20
C. 12
D. 25
Correct Answer: C

Solution:

Step 1: d^{2} = 17^{2} + 10^{2} - 2 (17) (10) cos 46° [Use law of cosines: d^{2} = e^{2}+ f^{2}- 2ef cos D.]
Step 2: d^{2} ~ 152.816154
Step 3: d = 12, to two significant digits. [Simplify.]