What formula is used when two sides and the included angle of a triangle are known?

When two sides and the included angle of a triangle are known, the appropriate formula to use is the Cosine Rule. This fundamental theorem in trigonometry relates the lengths of the sides of a triangle to the cosine of one of its angles. Specifically, if a triangle has sides of lengths a and b, with the included angle being C, the Cosine Rule states that:

c² = a² + b² - 2ab * cos(C)

Here, c represents the side opposite angle C. This formula is particularly useful for calculating the length of the third side of the triangle when the two sides and the angle between them are known. The Cosine Rule not only helps find side lengths but can also be rearranged to find angles when the side lengths are known, making it a versatile tool in triangle calculations.

In contrast, the other listed options do not apply to this situation: the Mean Formula pertains to averages and is unrelated to triangle calculations; Permutations deal with arrangements of objects and therefore have no relevance to triangles; and the Sine Rule is applicable when two angles and one side or two sides and a non-included angle are known. Thus, the Cosine Rule is clearly the correct choice for this scenario.