Using De Moivre's law to compute (−3+i)23Question:If z=−3+i, then z23=?

We can write z=r(cos⁡θ+isin⁡θ) z=2(cos⁡(56π)+isin⁡(56π)) z23=223(cos⁡(59π)+isin⁡(59π)) Now some trigonometry: cos⁡(59π)=cos⁡(12π+118π)=−sin⁡(118π) Similarly, sin⁡(59π)=cos⁡(118π), and you arrive to the answer.

Solved: "Using De Moivre's law to compute (−3+i)23Question:If z=−3+i,..."

High School
Geometry
Trigonometry
Trigonometric equations and identities

Alisha Pitts

Answered question

2022-01-24

Using De Moivre's law to compute

{(- \sqrt{3} + i)}^{\frac{2}{3}}

Question:
If

z = - \sqrt{3} + i

, then

z^{\frac{2}{3}} = ?

Answer & Explanation

ul2ph3ojc

Beginner2022-01-25Added 12 answers

We can write

z = r (\cos θ + i \sin θ)

z = 2 (\cos (\frac{5}{6} π) + i \sin (\frac{56}{π}))

z^{\frac{2}{3}} = 2^{\frac{2}{3}} (\cos (\frac{59}{π}) + i \sin (\frac{59}{π}))

Now some trigonometry:

\cos (\frac{59}{π}) = \cos (\frac{12}{π} + \frac{1}{18} π) = - \sin (\frac{1}{18} π)

Similarly,

\sin (\frac{59}{π}) = \cos (\frac{1}{18} π)

,
and you arrive to the answer.

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