#3**0 **

(a) By Law of Cosines,

\(c^2 = a^2 + b^2 - 2ab\cos \theta\\ a^2 - (2b\cos \theta)a+(b^2 - c^2) = 0\)

Now, we use the quadratic formula to get (try to fill in the steps on your own):

\(a = {b\cos \theta \pm \sqrt{ c^2 - b^2 \sin^2 \theta}}\)

(b) The condition where there are no solutions for a corresponds to the case that the expression inside the square root is negative, i.e. \(b\sin \theta > c\).

(c) The condition where there is 1 solution for a corresponds to the case that the square root equals 0 (since \(b\cos \theta \pm 0 = b\cos \theta\)), i.e., \(b\sin \theta = c\).

MaxWong Apr 17, 2022