Euler’s Formula


eix = cos(x) + isin(x) =cis(x)


~Note: cis(x) is short for cos(x) + isin(x)

~Note: cis(x) is the polar form of a complex number

Proof: The Taylor’s series expansions for ex, sin(x), and cos(x) at a=0 are used in the proof

ex = 1 + x+ (x2/2!) + (x3/3!) + (x4/4!) + …

sin(x) = x – (x3/3!) + (x5/5!) - …

cos(x) = 1 - (x2/2!) + (x4/4!) - …

Substituting ix for x in the expansion for ex , we get:

eix = 1 + (ix) + (ix)2/2! + (ix)3/3! + (ix)4/4! + …

Evaluating all powers of i and grouping the real and imaginary parts together, we get:

eix = (1 - x2/2! + x4/4! - …) + i(x -(x3/3!) + (x5/5!)…) = cos(x) + isin(x)

~Note: Euler’s formula is very useful for deriving most trig identities

Example: Deriving the sum formulas

ei(x+y) = cos(x+y) + isin(x+y) and ei(x+y) = (eix)(eiy)= [cos(x) + isin(x)] [cos(y) + isin(y)] = [cos(x)cosy –sin(x)sin(y)] + i[sin(x)cos(y) + cos(x)sin(y)], by multiplying and simplifying

Since two complex numbers are equal if and only if their real and imaginary parts are equal, we can equate the real parts to get:

(1) cos(x+y) = cos(x)cosy –sin(x)sin(y) and the imaginary parts to get:
(2) sin(x+y) = sin(x)cos(y) + cos(x)sin(y)

~Note: These are two of the most important identities in trig

From these two, we can easily derive the identity for (3) tan(x+y) by dividing (2) by (1)

Also, by substituting the appropriate expression for either x or y in (1), (2), and (3), we can easily derive the formulas for the double and half angles

For example: If we substitute x for y in (2), we get sin(2x)=2sin(x)cos(x)

Also, by substituting x for y in (1) along with the popular identity, sin2(x) + cos2(x) = 1, we can easily derive the popular identities sin2(x)=[1-cos(2x)]/2 and cos2(x)=[1+cos(2x)]/2 which are very useful in the integration process in calculus

~Note: If you substitue x=π into Euler's formula, you will get, eiπ=cos(π)+isin(π) or eiπ=1. Subtracting 1 from both sides gives, eiπ-1=0. This equation amazingly relates the 5 most important numbers in mathematics.

So, in conclusion, Euler’s formula is a great tool to have in your “mathematical bag”.