Law of Cosines

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A mathematical formula known as the law of cosines relates the lengths of the sides and the angles of a triangle. According to this rule, the square of a triangle's length is equal to the sum of the squares of its other two sides, minus twice the product of those two sides and the cosine of the angle between them. This relationship is called the "cosine formula."

The following form can be used to express the law of cosines:

c^2 = a^2 + b^2 - 2abcosC

where a, b, and c are the lengths of the triangle's sides, and C is the angle opposite the side of length c.

The law of cosines is often used to find missing side lengths or angles in a triangle when you know the lengths of two sides and the angle between them, or when you know the lengths of all three sides. It is a useful tool in trigonometry and geometry for solving problems involving triangles.

For instance, imagine you have a triangle with sides of lengths a = 3, b = 4, and c = 5, and angle C as you can see in the diagram. Using the law of cosines, you can find the measure of angle C like this:

cos C = (a^2 + b^2 - c^2) / (2ab)

= (9 + 16 - 25) / (2 * 3 * 4)

= -6 / 24

= -0.25

To obtain the measure of angle C, use a calculator or a table of cosines to obtain the value of cos C and then solve for C.

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