Question

If $a=\mathrm{cis}\alpha ,b=cis\beta ,c=cis\gamma$ then $\frac{a^3{b}^3}{c^2}=$

• A. $cis(3\alpha +3\beta +2\gamma )$
• B. $cis(3\alpha +3\beta -2\gamma )$
• C. $cis(-3\alpha -3\beta +2\gamma )$
• D. $cis(3\alpha -3\beta +2\gamma )$

Answer 1

The correct option is B $cis(3\alpha +3\beta -2\gamma )$

$a=cis\alpha =Cos\alpha +iSin\alpha =e^{i\alpha }$

$b=cis\beta =Cos\beta +iSin\beta =e^{i\beta }$

$a=cis\gamma =Cos\gamma +iSin\gamma =e^{i\gamma }$

$\therefore \frac{a^3{b}^3}{c^2}=\frac{\left(e^{i\alpha }{)}^3\right(e^{i\beta }{)}^3}{(e^{i\gamma }{)}^2}$

$=\frac{e^{3i\alpha }{e}^{3i\beta }}{e^{2i\gamma }}=e^{i(3\alpha +3\beta -2\gamma )}$

$=Cos(3\alpha +3\beta -2\gamma )+iSin(3\alpha +3\beta -2\gamma )$

$=cis(3\alpha +3\beta -2\gamma )$