If n is an integer and you replace θ by θ+2π in r^n(cos nθ+isin nθ), the value:

does not change, because 2nπ is a multiple of 2π

Exercise · ZW-DE-MOIVRE-FORMULA-009 · ★★★ Advanced

If \(n\) is an integer and you replace \(\theta\) by \(\theta+2\pi\) in \(r^n(\cos n\theta+i\sin n\theta)\), the value:

Choose one option.

Show answer

B. does not change, because \(2n\pi\) is a multiple of \(2\pi\)

Explanation

The new argument is \(n\theta+2n\pi\), and \(2n\pi\) is whole turns, so sine and cosine do not change: an integer power does not depend on the chosen argument.

Study Complex analysis in ZWMath →

Review the concept: De Moivre's formula

De Moivre's formula gives the integer power \(n\) of a complex number \(z\) of modulus \(r\) and argument \(\theta\): it raises the modulus to the \(n\)-th power and multiplies the argument by \(n\).

\[z^n=r^n\left(\cos n\theta+i\sin n\theta\right)\]

If \(n\) is an integer and you replace \(\theta\) by \(\theta+2\pi\) in \(r^n(\cos n\theta+i\sin n\theta)\), the value:

Explanation

Review the concept: De Moivre's formula

More practice problems: de moivre's formula