Why does e^ix = cos(x) + isin(x)

If you look at the Taylor series expansion of ex: ex = 1 + x + x2/(2!) + x3/(3!) + x4/(4!)...

If you then make this eix, you get

1+ix - x2/(2!) - ix3/(3!) + x4/(4!)...

If we split this into real and imaginary, we see the real part is

1 - x2/(2!) + x4/(4!) - x6/(6!)...

The series expansion of cos(x) is

1 - x2/(2!) + x4/(4!) - x6/(6!)...

Therefore, the real part of eix is cos(x)

If we look at only the imaginary part of eix, we get

i(x - x3/(3!) + x5/(5!) - x/ (7!)

If we look at the series expansion of sin(x) we get

(x - x3/(3!) + x5/(5!) - x/ (7!)

Therefore the imaginary part of eix = sin(x)

Putting this together, we get eix = Re(eix) + Im(eix)

eix = cos(x) + isin(x)

Jesse P. A Level Maths tutor, A Level Further Mathematics  tutor, GCS...

1 year ago

Answered by Jesse, an A Level Further Mathematics tutor with MyTutor

Still stuck? Get one-to-one help from a personally interviewed subject specialist


£26 /hr

Konrad E.

Degree: Natural Sciences: Physical (Masters) - Cambridge University

Subjects offered: Further Mathematics , Physics+ 2 more

Further Mathematics

“I'm studying Natural Sciences (Phys) at Cambridge. I enjoy tutoring science and maths at GCSE and A-Level, and sharing my experience.”

£20 /hr

Kyle W.

Degree: Computer Science (Bachelors) - Bristol University

Subjects offered: Further Mathematics , Maths

Further Mathematics

“About me Hi my name is Kyle! I'm currently in my second year studying Computer Science at the University of Bristol. I've always had a passion for mathematics and studying computer science has gave me an appreciation for just how many...”

£20 /hr

George I.

Degree: Electrical and Electronic Engineering (Masters) - Bristol University

Subjects offered: Further Mathematics , Maths

Further Mathematics

“About Me: I am currently a third year Electrical and Electronic engineering undergraduate at the University of Bristol. I have always enjoyed taking mathematics and further mathematics as I look forward to tackling problems with nume...”

MyTutor guarantee

About the author

Jesse P.

Currently unavailable: until 01/11/2015

Degree: Mathematics (Masters) - Oxford, New College University

Subjects offered: Further Mathematics , Maths+ 1 more

Further Mathematics

“About Me I'm a first year maths student at New College in Oxford. Maths has always felt like a game, rather than a subject to me, and I really hope I can show why, and get you to feel that way.  As well as the online tutoring, I tuto...”

MyTutor guarantee

You may also like...

Posts by Jesse

A table has diameter 130cm. What is the area, as a multiple of Pi

How does the product rule for differentiation work

When is the inequality x^4 < 8x^2 + 9 true?

Why does e^ix = cos(x) + isin(x)

Other A Level Further Mathematics questions

How do I find the asymptotes of a curve?

Understanding differentiation from first principle.

How do I do a proof by induction?

What is the complex conjugate?

View A Level Further Mathematics tutors


We use cookies to improve our service. By continuing to use this website, we'll assume that you're OK with this. Dismiss