Question

hi if i'm given complex numbers 2cis(pi/12) and 2cis(5pi/12) and represented with points a and b how do i find the distance between a and b?

Answer 1

\[\Large\bf\sf z_1=2cis\frac{\pi}{12},\qquad\qquad z_2=2cis\frac{5\pi}{12}\]
Ummm I think we just use the distance formula,\[\Large\bf\sf |z_1-z_2|\quad=\quad \sqrt{(x_1-x_2)^2+(y_1-y_2)^2}\]

Answer 2

Oh I called them z's instead of a and b, my bad :P

Answer 3

\[\large\bf\sf |a-b|\quad=\quad \sqrt{\left(2\cos\frac{\pi}{12}-2\cos\frac{5\pi}{12}\right)^2+\left(2\sin\frac{\pi}{12}-2\sin\frac{5\pi}{12}\right)^2}\]

Answer 4

help ur help @zepdrix
plz.. look

Answer 5

\[a=2\text{cis}\frac{\pi}{12}=2e^{i\pi/12}\\
b=2\text{cis}\frac{5\pi}{12}=2e^{5i\pi/12}\]
Then
\[\begin{align*}|a-b|&=\left|2e^{i\pi/12}-2e^{5i\pi/12}\right|\\
&=2\left|e^{i\pi/12}\left(1-e^{i\pi/3}\right)\right|\\
&=2\left|e^{i\pi}\right|^{1/12}\left|1-e^{i\pi/3}\right|\\
&=2\left|1-e^{i\pi/3}\right|\\
&=2\left|1-\left(\cos\frac{\pi}{3}+i\sin\frac{\pi}{3}\right)\right|\\
&=2\left|1-\left(\frac{1}{2}+i\frac{\sqrt3}{2}\right)\right|\\
&=2\left|\frac{1}{2}-i\frac{\sqrt3}{2}\right|\\
&=\left|1-i\sqrt3\right|
\end{align*}\]

Answer 6

...to get rid of the pesky \(\dfrac{\pi}{12}\).

Answer 7

thanks for your help