In triangle $ABC,$ $AB = 3,$ $AC = 5,$ $BC = 7,$ and $D$ lies on $\overline{BC}$ such that $\overline{AD}$ bisects $\angle BAC.$ Find $\cos \angle BAD.$
Law of Cosines
BC^2 = AC^2 + AB^2 - 2 (AC * AB) cos (BAC)
7^2 = 5^2 + 3^2 - 2 ( 5 * 3) cos (BAC)
[ 7^2 -5^2 -3^2] / [ -2 * 5 * 3] = cos (BAC) = -1/2
arccos (-1/2) = BAC = 120°
Since AD is a bisector ofthis angle, then BAD = 60°
cos 60° = cos (BAD) = 1/2