De Moivre's theorem

By using Maclaurin’s theorem, the power series for e^θ , sin θ and cos θ may be derived, and are: 1 https://www.w3.org/1998/Math/MathML"> e θ = 1 + θ + θ 2 2 ! + θ 3 3 ! + θ 4 4 ! + θ 5 5 ! + ⋯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315843629/20901bc5-1466-42ee-a7e4-0cc2fff34b68/content/math_370_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> 2 https://www.w3.org/1998/Math/MathML"> sin   θ = θ − θ 3 3 ! + θ 5 5 ! − θ 7 7 ! + ⋯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315843629/20901bc5-1466-42ee-a7e4-0cc2fff34b68/content/math_371_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> 3 https://www.w3.org/1998/Math/MathML"> and cos   θ = 1 − θ 2 2 ! + θ 4 4 ! − θ 6 6 ! + ⋯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315843629/20901bc5-1466-42ee-a7e4-0cc2fff34b68/content/math_372_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> Replacing θ by the imaginary number jθ in equation (1) gives: https://www.w3.org/1998/Math/MathML"> e j θ = 1 + j θ + ( j θ ) 2 2 ! + ( j θ ) 3 3 ! + ( j θ ) 4 4 ! + ( j θ ) 5 5 ! + ⋯ = 1 + j θ + j 2 θ 2 2 ! + j 3 θ 3 3 ! + j 4 θ 4 4 ! + j 5 θ 5 5 ! + ⋯ Since j 2 = − 1 , j 3 = − j , j 4 = 1  and  j 5 = j , then e j θ = 1 + j θ − θ 2 2 ! − j θ 3 3 ! + θ 4 4 ! + j θ 5 5 ! − ⋯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315843629/20901bc5-1466-42ee-a7e4-0cc2fff34b68/content/math_373_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> Grouping real and imaginary terms gives: 4 https://www.w3.org/1998/Math/MathML"> e j θ = ( 1 − θ 2 2 ! + θ 4 4 ! − ⋯ ) + j ( θ − θ 3 3 ! + θ 5 5 ! − ⋯ ) = cos   θ + j   sin   θ ,  from equations (2) and (3) above . Thus e j θ = c o s   θ + j   s i n   θ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315843629/20901bc5-1466-42ee-a7e4-0cc2fff34b68/content/math_374_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>