When you write out the rules you can see that multiplication and division are both involved:
Sine rule: a/sinA=b/sinB=c/sinC, so a/b=sinA/sinB, a/c=sinA/sinC, b/c=sinB/sinC; asinB=bsinA; asinC=csinA; bsinC=csinB.
Cosine rule: a^2=b^2+c^2-2bccosA; cosA=(b^2+c^2-a^2)/2bc; b^2=a^2+c^2-2accosB; cosB=(a^2+c^2-b^2)/2ac; c^2=a^2+b^2-2abcosC; cosC=(a^2+b^2-c^2)/2ab.