Question

Use de moivre's theorem to write the complex number in trigonometric form (cos(2pi/7)+isin(2pi/7))^5

Answer 1

By DeMoivre's theorem, you have


`\left(\cos\frac{2\pi}7+i\sin\frac{2\pi}7\right)^5=\cos\frac{10\pi}7+i\sin\frac{10\pi}7`

Answer 2

`(\cos{(2\pi)/(7)}+i\sin{(2\pi)/(7)})^5 =\cos{((10\pi)/(7))}+i\sin{((10\pi)/(7))}`

Explanation:

Given the complex number

`(\cos{(2\pi)/(7)}+i\sin{(2\pi)/(7)})^5`

we have to write the complex number in trigonometric form using de moivre's theorem.

By de Moivre's formula,

`(\cos x+i \sin x)^ n=cos(nx)+isin(nx)`

∴
`(\cos{(2\pi)/(7)}+i\sin{(2\pi)/(7)})^5=\cos{5((2\pi)/(7))}+i\sin{5((2\pi)/(7))}`

`=\cos{((10\pi)/(7))}+i\sin{((10\pi)/(7))}`

which is required form