anonymous
  • anonymous
Find the fourth roots of the complex number z1= 1+ sqrt3 I Part 1:write z1 in polar form 2(cos60+isin60) Part 2: find the modulus of the roots of z1 I got 2 Part 3: find the four angles that define the fourth roots of the number z1 Part 4: what are the fourth roots of z1= sqrt3+1 i
Mathematics
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anonymous
  • anonymous
Find the fourth roots of the complex number z1= 1+ sqrt3 I Part 1:write z1 in polar form 2(cos60+isin60) Part 2: find the modulus of the roots of z1 I got 2 Part 3: find the four angles that define the fourth roots of the number z1 Part 4: what are the fourth roots of z1= sqrt3+1 i
Mathematics
katieb
  • katieb
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anonymous
  • anonymous
am I correct on part 1&2 and I don't know how to do 3&4
IrishBoy123
  • IrishBoy123
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anonymous
  • anonymous

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anonymous
  • anonymous
If \(z\) has angle \(\theta\), then the \(n\)th roots \(z^{1/n}\) will follow a pattern of \(\dfrac{\theta+2k\pi}{n}\), where \(k=0,1,\ldots,n-1\).
anonymous
  • anonymous
@SithsAndGiggles so if the angles i find are 15 and 60 the 4th roots would be \[60+2k \div4\]?
anonymous
  • anonymous
Idk i just have no idea how to do this
anonymous
  • anonymous
You found that \(\theta=60^\circ\), right? In radians, that's \(\dfrac{\pi}{3}\). Take \(k=0\). Then the angle of the first (principal) fourth root is \(\dfrac{\dfrac{\pi}{3}+2\pi\times0}{4}=\dfrac{\pi}{12}\), which in degrees is \(15^\circ\). Now take \(k=1\). This gives you an angle of \(\dfrac{\dfrac{\pi}{3}+2\pi\times1}{4}=\dfrac{7\pi}{12}\), or \(105^\circ\). Continue the pattern for \(k=2\) and \(k=3\).

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