Question

The cosine law

Answer 1

$\bf \textit{Law of sines} \\ \quad \\ \cfrac{sin(\measuredangle A)}{a}=\cfrac{sin(\measuredangle B)}{b}=\cfrac{sin(\measuredangle C)}{c}\\\\ -------------------------------$

$\bf \cfrac{sin(88^o)}{39.9}=\cfrac{sin(A)}{31}\implies \cfrac{31\cdot sin(88^o)}{39.9}=sin(A) \\\\\\ sin^{-1}\left[ \cfrac{31\cdot sin(88^o)}{39.9} \right]=sin^{-1}\left[ sin(A) \right] \\\\\\ sin^{-1}\left[ \cfrac{31\cdot sin(88^o)}{39.9} \right]=\measuredangle A \\\\\\ 50.93841^o\approx \measuredangle A\qquad thus\qquad \measuredangle C\approx 41.0619^o$