Problem: \(z^6 + z^3 + 1 = 0\) has one root whose argument is between \(\frac{\pi}{2}\) and \(\pi\). What is the argument?
Solution:
\(z^6 + z^3 + 1 = 0\)
\(\Rightarrow (z^3 - 1)(z^6 + z^3 + 1) = 0\)
\(\Rightarrow (z^9 - 1) = 0\)
\(\Rightarrow z = \cos\frac {2k\pi}{9} + \sin\frac{2k\pi}{9}\)
Required argument is found by putting values for k (=3)
Solution:
\(z^6 + z^3 + 1 = 0\)
\(\Rightarrow (z^3 - 1)(z^6 + z^3 + 1) = 0\)
\(\Rightarrow (z^9 - 1) = 0\)
\(\Rightarrow z = \cos\frac {2k\pi}{9} + \sin\frac{2k\pi}{9}\)
Required argument is found by putting values for k (=3)
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