find all six 6throot of unity and write them in rectangular form.check numerically thatώ+ώ^2+ώ^3 ώ^4+ώ^5 =-1?
Please someone help i am failing my class with these problems. Help. (ignore the commor over the omaga).due wednesday 22nd 2009 Thanks in advance.
The easiest way to find the sixth root of unity is to convert to polar
form then use DeMoivre’s theorem.
Unity is 1 = 1 + 0i in Cartesian Form and in polar form it is
just
1 cos 0° + i sin 0° . We abbreviate that 1 cis 0°
but 0° also correponds to 360° on the polar plane
so we can call unity
1 cis 360°
from DeMoiveres theorm, we know the first 6th root is equal
to 1^(1/6) cis (360°/6) and
1^(1/6) is simply 1. So first 6th root is
1 cis 60°
The other five roots are found by rotating 1/6 of a revolution
around the polar plane to find yourself at
1 cis 120°
1 cis 180°
1 cis 240°
1 cis 300°
Those are the roots in polar form. You now have to convert each
to rectangular form
1 cis 60° is
short for 1 cos 60° + i sin 60°
which is √3/2 + (1/2)i
Next
1 cis 120° is
short for 1 cos 120° + i sin 120°
which is -√3/2 + (1/2)i
1 cis 180°
is
short for 1 cos 180° + i sin 180°
which is -1 + 0i and that is just -1
I’ll let you do 240° and 300° by your self
the second part is a large binomial expansion. You may need
a page of paper but it is substitution, collecting like terms and
simplifying, and making sure you know i² = -1
i^3 = -i i^4 = 1 and i^5 = i
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