Complex Numbers (1 Viewer)

Vision · Dec 8, 2013

Hey,

Simplify, leaving your answers in exact values.

$\left(\frac{1+cos\frac{\pi}{7}+isin\frac{\pi}{7}}{1+cos\frac{\pi}{7}-isin\frac{\pi}{7}}\right)^7$

Thanks in advance

Carrotsticks · Dec 8, 2013

Split cos(pi/7) as cos(2*pi/14) and use the appropriate double angle formula to eliminate the '1'.

Same for the sine, and it reduces to cis(something)/cis(something), which can be evaluated easily.

anomalousdecay · Dec 8, 2013

I remember doing this question exactly one year ago in Terry Lee's book I think. If I find my solution then I will post it up.

Vision · Dec 8, 2013

anomalousdecay said:
I remember doing this question exactly one year ago in Terry Lee's book I think. If I find my solution then I will post it up.

Haha yeah it was from Terry Lee. Don't worry about posing the solution though, I managed to figure it out (Thanks carrot).

Drongoski · Dec 8, 2013

$= \left (\frac {2cos^2 \frac{\pi}{14} +2isin \frac{\pi}{14} cos \frac {\pi}{14}}{2cos^2 \frac {\pi}{14} - 2i sin \frac{\pi}{14} cos \frac {\pi}{14}} \right )^7 = \left( \frac {2cos \frac {\pi}{14} (cos \frac {\pi}{14} + isin \frac {\pi}{14})}{2cos \frac {\pi}{14} (cos \frac {\pi}{14} - isin \frac {\pi}{14})} \right)^7 = \left (\frac { cis \frac {\pi}{14}}{cis \frac {-\pi}{14}}\right )^7\\ \\ = \left ( \frac {cis \frac {\pi}{2}}{cis \frac {-\pi}{2}}\right ) = \frac {i}{-i} = -1$

Hope that's correct.

Edit: have used following identities:

$2cos^2 \theta = 1 + cos 2\theta \\ \\ sin 2\theta = 2sin \theta cos\theta$

timeflies · Dec 8, 2013

Drongoski said:
$= \left (\frac {2cos^2 \frac{\pi}{14} +2isin \frac{\pi}{14} cos \frac {\pi}{14}}{2cos^2 \frac {\pi}{14} - 2i sin \frac{\pi}{14} cos \frac {\pi}{14}} \right )^7$

Wait... how did you get to there from the question? I'm a little lost :/

rumbleroar · Dec 8, 2013

timeflies said:
Wait... how did you get to there from the question? I'm a little lost :/

You use double angle formula. I'll post up a picture.

rumbleroar · Dec 8, 2013

ImageUploadedByTapatalk1386501843.169852.jpg

Let me know if it makes sense/doesn't make sense!

Apologies for off handwriting, hand is dead from MX1 study haha

Sent from my iPhone using Tapatalk

timeflies · Dec 8, 2013

rumbleroar said:
View attachment 29342

Let me know if it makes sense/doesn't make sense!

Apologies for off handwriting, hand is dead from MX1 study haha

Sent from my iPhone using Tapatalk

Ohh I get it now! Thanks a lot

Complex Numbers (1 Viewer)

Vision

New Member

Carrotsticks

Retired

anomalousdecay

Premium Member

Vision

New Member

Drongoski

Well-Known Member

timeflies

Active Member

rumbleroar

Survivor of the HSC

rumbleroar

Survivor of the HSC

timeflies

Active Member

Users Who Are Viewing This Thread (Users: 0, Guests: 1)