What is the formula for finding the cosine of angle C in a triangle?

The formula for finding the cosine of angle C in a triangle is derived from the Law of Cosines, which states that for any triangle with sides a, b, and c, the relation connecting the sides to the cosine of one of its angles can be expressed as:

[

c^2 = a^2 + b^2 - 2ab \cos C

]

Rearranging this formula to solve for ( \cos C ) gives us:

[

\cos C = \frac{a^2 + b^2 - c^2}{2ab}

]

Thus, the correct representation of the cosine of angle C is:

[

\cos C = \frac{a^2 + b^2 - c^2}{2ab}

]

This shows that the formula appropriately incorporates the lengths of the triangle's sides in a way that reflects the cosine relationship. It effectively establishes how the length of the sides a and b in conjunction with the opposing side c determines the cosine of angle C.