Expand Using De Moivre's Theorem (2pi)/3

Problem

(cos((2*π)/3)+i*sin((2*π)/3))n

Solution

1. Identify the complex number in polar form z=cos(θ)+i*sin(θ) where θ=(2*π)/3

2. Apply De Moivre's Theorem, which states that for any integer n (cos(θ)+i*sin(θ))n=cos(n*θ)+i*sin(n*θ)

3. Substitute the given angle θ=(2*π)/3 into the theorem formula.

4. Simplify the expression by multiplying the angle by n to obtain the expanded form.

Final Answer

(cos((2*π)/3)+i*sin((2*π)/3))n=cos((2*n*π)/3)+i*sin((2*n*π)/3)

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