3 Use DeMoivres theorem to evaluate the expression Write the

3. Use DeMoivre\'s theorem to evaluate the expression Write the result in standard form. (2(cos 36 +isin 36))

Solution

By demoivre\'s theorem,

e^ix = cosx + i sinx

Here taking x = 36

cos36 + isin36 = eⁱ⁽³⁶⁾

^then,

(2(cos36 + isin36))⁵ = 2⁵(cos36 + isin36)⁵

= 32(eⁱ⁽³⁶⁾)⁵

= 32(eⁱ⁽¹⁸⁰⁾)

= 32(-1)

= -32

{ eⁱ⁽¹⁸⁰⁾ = cos180 + isin180

= -1 + 0

= -1. }