# In order to apply the law of cosines to find the length of the side of a triangle, it is enough to know which of the following ?the area of a triangle ?

The measure of an angle and the length of the side opposite that angle ?

the lengths of the two triangle sides and the measure of the angle between them ?

**Solution:**

In order to apply the law of cosines to find the length of the side of a triangle, we need to know the length of two sides of the triangle and the measure of the angle.

In trigonometry, the law of cosines relates the lengths of the sides of a triangle to the cosine of one of its angles.

As in figure below, the law of cosines states

c^{2} = a^{2} + b^{2} - 2ab cosγ

Also it can be written as:

a^{2} = b^{2} + c^{2} - 2bc cos⍺

b^{2} = c^{2} + a^{2} - 2ca cosꞵ

The law of cosines generalizes the Pythagorean theorem, which holds only for right triangles:

if the angle γ is a right angle (of measure 90 degrees, or π/2 radians), then cos γ = 0, and thus the law of cosines reduces to the Pythagorean theorem:

c^{2} = a^{2} + b^{2}

For example: Given a triangle with the three sides: a = 8, b = 6 and c = 9. Find angle C.

cos C = (a^{2} + b^{2} − c^{2})/2ab

cos C = (8^{2} + 6^{2} − 9^{2})/(2×8×6)

cos C = (64 + 36 − 81)/96

cos C = 19/96 = 0.19791

C = cos^{−1}(0.19791) = 78.58°

## In order to apply the law of cosines to find the length of the side of a triangle, it is enough to know which of the following ?

the area of a triangle ?

The measure of an angle and the length of the side opposite that angle ?

the lengths of the two triangle sides and the measure of the angle between them ?

**Summary:**

In order to apply the law of cosines to find the length of the side of a triangle, we need to know the length of two sides of the triangle and the measure of the angle.

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