how do you use the law of cosines to find the missing side length of triangle ABC?

Answer 1

The law of cosines is c^2 = a^2 + b^2 - 2ab*cos(C), where c is the side opposite angle C, and so on. So in order to get the missing side length, you will need to know the measure of one of the angles in the triangle. If you know the measures of two sides and one angle, you can formulate the law of cosines with one unknown. You can use the sides a, b, and c in any order; for example, a^2 = b^2 + c^2 - 2bc*cos(A) is a valid formulation as well.

Answer 2

If the angle's measure is known that is opposite the remaining unknown side, it is then merely a case of assigning C to the unknown side and c to the opposite angle, assigning a and b to the remaining (and known) sides, and solving for C², finally taking the square root.

If the known angle is opposite one of the known sides, you can rearrange the equation as

x² - 2Bx*cos c + (B² - C²) = 0

where the unknown side is relabeled x, the side opposite the known angle is C, the known angle is c, and the remaining known side is B, then solving this as a quadratic equation in x. The solution(s) for x will be
x = Bcos c ± √[C² - B²(1 - cos² c)]
If desired, this can also be written as
x = Bcos c ± √C² - B²sin² c)
Of course, only positive solutions for x would be admissible Two nondegenerate solutions are possible only if c is an acute angle and B is the longer of the known sides. If B is the hypotenuse of a right triangle, the two solutions of x will be the same.

Answer 3

if a is missing and b, c are given in addition to A, then
a = √ [b^2 + c^2 -- 2bccosA]