**Definition of Law of Cosines**

- 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}- 2*bc*cos*A*

*b*^{2}=*a*^{2}+ c^{2}- 2*ac*cos*B*

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

**Example of Law of Cosines**

- 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^{2}+ c^{2}-*bc*cos*A*

Cos*A*=*b*^{2}+ c^{2}- a^{2}/ 2*bc*

Cos*A*= 12^{2}+ 10^{2}- 19^{2}/ 2(12) (10)

Cos*A*= - 0.4875

∠*A*= 119°

**Solved Example on Law of Cosines**

In ΔDEF, if angle

D= 46°,f= 10, ande= 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}= ef^{2}+2ef cos^{2}-D.]

Step 2:d˜ 152.816154^{2}

Step 3:d= 12, to two significant digits. [Simplify.]

**Related Terms for Law of Cosines**

- Angle
- Cos
- Equation
- Law of Sines
- Length
- Side
- Triangle