cis(x) = cos(x) + isin(x)
cis(11pi/6) = cos(11pi/6) + isin(11pi/6)
cos(11pi/6) = √3/2
sin(11pi/6) = 1/2
So the top then becomes:
6√5∗(√3+i)/2
This expands to:
(6√15+6i√5)/2
Now we do the same with the denominator.
cis(pi/2) = cos(pi/2) + isin(pi/2)
cos(pi/2) = 0
sin(pi/2) = 1
cis(pi/2) = 0 + i
Our denominator then becomes:
3√6∗0+3i√6
Our answer is then:
((6√15+6i√5)/2)/3i√6 which ends up simplifying to:
(6√15+6i√5)/6i√6
Rationalizing our denominator, we get:
(6√90+6i√30)/36i=(3√10+i√30)/6i
6√5 cis(11pi/6) ÷ 3√6 cis(pi/2)
6√5e(11πi/6)÷3√6e(πi/2) =2√5e[(11πi/6)−(πi/2)]÷√6 =2√5e[(11πi/6)−(3iπ/6)]÷√6 =2√5e[(4πi/3)]÷√6 =2√30[cos(4πi/3)+isin(4πi/3)]÷6 =√303[−cos(π/3)−isin(π/3)] =√303[−12−i(√32)] =√306[−1−√3i]
I think this is the same as jfan's answer,.
jfan, you need to finish by making your denominator real.
I will multiply mine out as well, it might look better.....
=√306[−1−√3i] =−16[√30+3√10i] =−√106[√3+3i]
Does that look better? Not sure.
LaTex:
Latex:
6\sqrt5\;e^{(11\pi i/6)}\div 3\sqrt6 \;e^{(\pi i/2)}\\~\\
=2\sqrt5\;e^{[(11\pi i/6)-(\pi i/2)]}\div \sqrt6 \\~\\
=2\sqrt5\;e^{[(11\pi i/6)-(3i\pi/6)]}\div \sqrt6 \\~\\
=2\sqrt5\;e^{[(4\pi i/3)]}\div \sqrt6 \\~\\
=2\sqrt{30}[\;cos(4\pi i/3)+isin(4\pi i/3)]\div 6 \\~\\
=\frac{\sqrt{30}}{3}[\;-cos(\pi /3)-isin(\pi /3)] \\~\\
=\frac{\sqrt{30}}{3}[\;-\frac{1}{2}-i(\frac{\sqrt3}{2})] \\~\\
=\frac{\sqrt{30}}{6}[\;-1-\sqrt3i] \\~\\