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6√5 cis(11pi/6) ÷ 3√6 cis(pi/2)

 Apr 25, 2020
 #1
avatar+499 
+1

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:

65(3+i)/2

This expands to:

(615+6i5)/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:

360+3i6

Our answer is then:

((615+6i5)/2)/3i6 which ends up simplifying to:

(615+6i5)/6i6

Rationalizing our denominator, we get:

(690+6i30)/36i=(310+i30)/6i

 Apr 25, 2020
edited by jfan17  Apr 25, 2020
edited by jfan17  Apr 25, 2020
edited by jfan17  Apr 25, 2020
edited by jfan17  Apr 25, 2020
 #2
avatar+118696 
+1

6√5 cis(11pi/6) ÷ 3√6 cis(pi/2)

 

 

65e(11πi/6)÷36e(πi/2) =25e[(11πi/6)(πi/2)]÷6 =25e[(11πi/6)(3iπ/6)]÷6 =25e[(4πi/3)]÷6 =230[cos(4πi/3)+isin(4πi/3)]÷6 =303[cos(π/3)isin(π/3)] =303[12i(32)] =306[13i] 

 

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[13i] =16[30+310i] =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] \\~\\

 Apr 25, 2020
edited by Melody  Apr 25, 2020

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