Toolkit 107Law of SinesasinA=bsinB=csinC\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}sinAa=sinBb=sinCcLaw of SinesasinA=bsinB=csinC\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}sinAa=sinBb=sinCcProofUsing the area formula of triangle ABC,[ABC]=12acsinB=12bcsinA.[ABC]=\frac12 ac\sin B=\frac12 bc\sin A.[ABC]=21acsinB=21bcsinA.Therefore,asinB=bsinA.a\sin B=b\sin A.asinB=bsinA.asinA=bsinB.\frac{a}{\sin A}=\frac{b}{\sin B}.sinAa=sinBb.Similarly,asinA=bsinB=csinC.□\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}.\quad\squaresinAa=sinBb=sinCc.□
