Find the exponential form of the complex number
[e^{\pi i/4} + e^{5 \pi i/12} + e^{7 \pi i/12} + e^{3\pi i/4} + e^{11\pi i/12}
\(e^{\pi i/4} + e^{5 \pi i/12} + e^{7 \pi i/12} + e^{3\pi i/4} + e^{11\pi i/12}\\ \)
with proof.
I've graphed each of the points on the complex plane to find that they are separated by an angle measure of pi/6. 2 pairs of 2 points have the same x-coordinate, but negative, but I don't know where to go from there. I know I'm supposed to use the magnitude of the sum of some of the points as well as the symmetry of the diagram, but I'm not sure how.
Thanks!
Use the identity
\(\displaystyle re^{\imath \theta}\equiv r(\cos \theta + \imath \sin \theta)\)
to switch each term to trig and then use the identities
\(\displaystyle \cos A + \cos B = 2\cos((A +B)/2)\cos((A-B)/2) \\\sin A +\sin B = 2\sin((A+B)/2)\cos((A-B)/2)\)
to group terms, (1st with 5th, 2nd with 4th).
Switch back to exponential form at the end.
+1712 I have no idea what the answer is but for convenience's sake:
THE QUESTION:
Find the exponential form of the complex number
\( [e^{\pi i/4} + e^{5 \pi i/12} + e^{7 \pi i/12} + e^{3\pi i/4} + e^{11\pi i/12} \)
with proof.
Use the identity
\(\displaystyle re^{\imath \theta}\equiv r(\cos \theta + \imath \sin \theta)\)
to switch each term to trig and then use the identities
\(\displaystyle \cos A + \cos B = 2\cos((A +B)/2)\cos((A-B)/2) \\\sin A +\sin B = 2\sin((A+B)/2)\cos((A-B)/2)\)
to group terms, (1st with 5th, 2nd with 4th).
Switch back to exponential form at the end.
