G8.1 If $A=\frac{5 \sin \theta+4 \cos \theta}{3 \sin \theta+\cos \theta}$ and $\tan \theta=2$, find $A$.
G8.2 If $x+\frac{1}{x}=2 A$, and $x^{3}+\frac{1}{x^{3}}=B$, find $B$.
$\begin{aligned} A & =\frac{(5 \sin \theta+4 \cos \theta) \div \cos \theta}{(3 \sin \theta+\cos \theta) \div \cos \theta} \\ & =\frac{5 \tan \theta+4}{3 \tan \theta+1} \\ & =\frac{5(2)+4}{3(2)+1}=2 \\ x & +\frac{1}{x}=4 \\ \Rightarrow & x^{2}+\frac{1}{x^{2}}=4^{2}-2=14 \\ B & =x^{3}+\frac{1}{x^{3}} \\ & =\left(x+\frac{1}{x}\right)\left(x^{2}-1+\frac{1}{x^{2}}\right) \\ & =4(14-1)=52\end{aligned}$