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Exponential Form and Euler’s Formula

Key Terms

Euler’s formula
eiθ = cosθ + i sinθ for any real θ.
Exponential form
Any complex number z = r(cosθ + i sinθ) can be written as z = reiθ, where r = |z| and θ = arg(z).
Euler’s identity
Setting θ = π gives eiπ + 1 = 0 — connecting e, i, π, 1 and 0.
Multiplication
r1eiθ1 × r2eiθ2 = r1r2ei(θ1+θ2).
Division
(r1eiθ1) ÷ (r2eiθ2) = (r1/r2)ei(θ1−θ2).
De Moivre in exponential form
(reiθ)n = rneinθ.
Conjugate
If z = reiθ, then z̅ = re−iθ.
Key Identities from Euler’s Formula:
  • cosθ = (eiθ + e−iθ) / 2
  • sinθ = (eiθ − e−iθ) / (2i)
  • eiπ + 1 = 0   (Euler’s identity)
  • eiπ/2 = i,   eiπ = −1,   ei⋅2π = 1
Hot Tip When converting between polar (cis) form and exponential form: r cisθ = reiθ — they are identical representations. In exponential form, multiplication becomes ordinary index arithmetic: add angles, multiply moduli. This makes De Moivre’s theorem completely transparent: (reiθ)n = rneinθ follows directly from index laws.

Worked Example 1 — Convert 2cis(π/3) to exponential form

We have z = 2(cos(π/3) + i sin(π/3)) = 2 cis(π/3).

By Euler’s formula, eiθ = cosθ + i sinθ, so:

z = 2eiπ/3

To find Cartesian form: z = 2(cos(π/3) + i sin(π/3)) = 2(1/2 + i√3/2) = 1 + i√3.

Worked Example 2 — Multiply z₁z₂ in exponential form

Let z1 = 3eiπ/4 and z2 = 2eiπ/3.

z1z2 = 3 × 2 × ei(π/4 + π/3) = 6ei(3π/12 + 4π/12) = 6ei7π/12.

Modulus = 6, argument = 7π/12 (105°).

Worked Example 3 — Prove cosθ = (eiθ + e−iθ)/2

By Euler’s formula: eiθ = cosθ + i sinθ  … (1)

Replace θ with −θ: e−iθ = cos(−θ) + i sin(−θ) = cosθ − i sinθ  … (2)

Adding (1) and (2): eiθ + e−iθ = 2cosθ

Therefore cosθ = (eiθ + e−iθ)/2. ✓

Subtracting (2) from (1): eiθ − e−iθ = 2i sinθ, giving sinθ = (eiθ − e−iθ)/(2i).

Why Euler’s Formula Is True

Euler’s formula eiθ = cosθ + i sinθ can be motivated by comparing the Taylor series of each side. Recall that for a real number x:

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

Substitute x = iθ:

eiθ = 1 + iθ + (iθ)2/2! + (iθ)3/3! + (iθ)4/4! + …

= 1 + iθ − θ2/2! − iθ3/3! + θ4/4! + iθ5/5! − …

Separating real and imaginary parts:

Real: 1 − θ2/2! + θ4/4! − … = cosθ

Imaginary: θ − θ3/3! + θ5/5! − … = sinθ

So eiθ = cosθ + i sinθ. ✓

The Exponential Form z = reiθ

Every non-zero complex number can be written in the exponential form z = reiθ, where r = |z| > 0 is the modulus and θ = arg(z) is the argument. This form makes multiplication, division, and powers particularly elegant.

Converting from Cartesian to exponential: Given z = a + bi, first find r = √(a²+b²) and θ = arg(z) = atan2(b, a). Then z = reiθ.

Converting from exponential to Cartesian: Given z = reiθ, expand using Euler’s formula: z = r(cosθ + i sinθ) = r cosθ + ir sinθ.

Algebra in Exponential Form

Multiplication: If z1 = r1eiθ1 and z2 = r2eiθ2:

z1z2 = r1r2 ei(θ1+θ2)

Geometrically: multiply moduli, add arguments.

Division:

z1/z2 = (r1/r2) ei(θ1−θ2)

Powers (De Moivre): (reiθ)n = rneinθ for any integer n. This is simply the index law (am)n = amn.

Euler’s Identity: eiπ + 1 = 0

Setting θ = π in Euler’s formula: eiπ = cosπ + i sinπ = −1 + 0 = −1.

Rearranging: eiπ + 1 = 0.

This equation combines the five most fundamental constants in mathematics: e (base of natural logarithm), i (imaginary unit), π (ratio of circumference to diameter), 1 (multiplicative identity), and 0 (additive identity). It is frequently voted the most beautiful equation in mathematics.

Expressing Trig Functions via Exponentials

From eiθ = cosθ + i sinθ and e−iθ = cosθ − i sinθ:

  • cosθ = (eiθ + e−iθ)/2
  • sinθ = (eiθ − e−iθ)/(2i)

These identities are analogous to the definitions of hyperbolic functions: cosh(x) = (ex + e−x)/2 and sinh(x) = (ex − e−x)/2. The relationship cosθ = cosh(iθ) connects circular and hyperbolic trigonometry.

Exam Tip: r cisθ and reiθ are identical — both mean the same complex number. The exponential form makes index laws crystal clear: to multiply, add exponents; to divide, subtract exponents; to raise to a power, multiply the exponent.
Exam Tip: When a question asks you to “use Euler’s formula,” show the step eiθ = cosθ + i sinθ explicitly before substituting. Do not just write the answer.

Mastery Practice

  1. Convert between forms. Fluency

    Write z = 3eiπ/4 in polar form r cisθ, then find the real and imaginary parts of z.

  2. Convert to exponential form. Fluency

    Write z = 2(cos(π/6) + i sin(π/6)) in exponential form.

  3. Multiply in exponential form. Fluency

    If z1 = 2eiπ/3 and z2 = 3eiπ/6, find z1z2 in exponential form, then convert to a + bi.

  4. Divide in exponential form. Fluency

    Find z1/z2 if z1 = 4ei2π/3 and z2 = 2eiπ/6. Leave your answer in exponential form.

  5. Euler’s identity. Understanding

    Use Euler’s formula to show that eiπ + 1 = 0. Explain in one or two sentences why this equation is described as “the most beautiful equation in mathematics.”

  6. Multiple angle formulas. Understanding

    Use the fact that ei3θ = (eiθ)3 to expand and find expressions for cos(3θ) and sin(3θ) in terms of cosθ and sinθ. Show all working.

  7. Modulus and argument of a complex exponential. Understanding

    Find |z| and arg(z) for z = e2+iπ/4, given that e2+iπ/4 = e2 ⋅ eiπ/4. Express the argument in exact form.

  8. Verify De Moivre’s theorem. Understanding

    Show that (eiθ)n = einθ using index laws, and hence verify De Moivre’s theorem for z = r cisθ: that zn = rn cis(nθ).

  9. Power reduction using Euler’s formula. Problem Solving

    Using the identities cosθ = (eiθ + e−iθ)/2 and sinθ = (eiθ − e−iθ)/(2i), find an expression for cos2θ in terms of cos(2θ). Show all steps.

  10. Unit circle and Euler’s formula. Problem Solving

    A complex number z satisfies |z| = 1. Write z = eiθ and find all values of θ in [0, 2π) such that z + z̅ = √3. Express answers in exact form.

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