Euler's formula — e^(iθ) = cos θ + i sin θ (proof via Taylor series): Question bank

The words we lean on (all built in the parent note): the modulus $∣ z ∣$ is the distance of the point $z$ from the origin; the argument is the angle it makes with the positive real axis; radians measure angle by arc-length on the unit circle ( $π$ rad $= 180°$ ); a function is entire if its power series converges for every complex input.

True or false — justify

TF1.

e^{i θ}

grows without bound as

θ

increases, just like

e^{x}

for real

x

False. The exponent is imaginary, so

∣ e^{i θ} ∣ = cos^{2} θ + sin^{2} θ = 1

forever — increasing

θ

spins the point around the unit circle, it never leaves it.

TF2. Euler's formula

e^{i θ} = cos θ + i sin θ

holds only for

0 \leq θ < 2 π

False. It holds for every real

θ

, positive, negative, or huge. Adding

2 π

just returns to the same point since

cos, sin

have period

2 π

; the formula itself has no restricted domain.

TF3.

e^{i θ}

and

e^{i (θ + 2 π)}

are the same complex number.

True. Both equal

cos θ + i sin θ

because cosine and sine repeat every

2 π

. They are the same point, reached after one extra full turn.

TF4.

\overline{e^{i θ}} = e^{- i θ}

(bar = complex conjugate).

True. Conjugating flips the sign of the imaginary part:

\overline{cos θ + i sin θ} = cos θ - i sin θ = cos (- θ) + i sin (- θ) = e^{- i θ}

TF5. Since

∣ e^{i θ} ∣ = 1

, the number

e^{i θ}

can never be a real number.

False. At

θ = 0

it is

1

and at

θ = π

it is

- 1

— both real. It is real exactly when

sin θ = 0

, i.e.

θ

is a multiple of

π

TF6. For

θ = π

, Euler's formula gives

e^{iπ} = - 1

, and this is consistent with a half-turn on the circle.

True. A half-turn (

π

rad

= 180°

) sends the tip from

+ 1

- 1

. This is Euler's identity in disguise.

TF7. The identity

(cos θ + i sin θ)^{n} = cos n θ + i sin n θ

is true only for positive integer

n

False. Via

e^{i θ}

it holds for any integer

n

(including negative), since

(e^{i θ})^{n} = e^{in θ}

for all integers. (See De Moivre's Theorem for the extension to rationals.)

TF8.

e^{i θ} \cdot e^{i ϕ} = e^{i (θ + ϕ)}

is just the ordinary exponential law

a^{x} a^{y} = a^{x + y}

, and geometrically it means angles add.

True. Multiplying two unit-circle points adds their angles — that is exactly rotation stacking, and the exponential law is what encodes it algebraically.

Spot the error

SE1. "Compute

e^{i \cdot 90}

for a right angle:

cos 90 + i sin 90 \approx - 0.45 + 0.89 i

The angle must be in radians. A right angle is

θ = π /2

, giving

e^{iπ /2} = i

. The number

90

radians is a totally different point; the Taylor-series proof only produces

cos, sin

for radian arguments.

SE2. "

sin θ = \frac{e ^{i θ} - e ^{- i θ}}{2}

, by symmetry with the cosine formula."

Subtracting gives

(cos θ + i sin θ) - (cos θ - i sin θ) = 2 i sin θ

, so you must divide by

2i

, not

2

. Correct:

sin θ = \frac{e ^{i θ} - e ^{- i θ}}{2 i}

SE3. "

(cos θ + i sin θ)^{3} = cos θ^{3} + i sin θ^{3}

De Moivre multiplies the angle, not the trig value: the answer is

cos 3 θ + i sin 3 θ

. You cannot distribute a power across a sum like this.

SE4. "Proof step: substitute

x = i θ

, then rearrange the series into real and imaginary groups — allowed for any series."

Rearrangement is only legal because the series is absolutely convergent (which it is, everywhere, since

e^{x}

is entire). For a merely conditionally convergent series, reordering can change the sum.

SE5. "

e^{i θ} = cos θ + i sin θ

, so the real part of

e^{2 i θ}

cos^{2} θ

The real part of

e^{2 i θ}

cos 2 θ

, not

cos^{2} θ

. (They're related:

cos 2 θ = 2 cos^{2} θ - 1

, but they are not equal.)

SE6. "Since

cos θ = \frac{e ^{i θ} + e ^{- i θ}}{2}

is a sum of two things of modulus 1, its value can reach 2."

The two terms are conjugates that point in opposite vertical directions; their sum is real and lies in

[- 2, 2]

before dividing, so

cos θ \in [- 1, 1]

as required. Modulus-1 pieces do not simply add in magnitude.

SE7. "In the proof, the imaginary bracket

θ - \frac{θ ^{3}}{3 !} + \dots

is the series for

cos θ

because it alternates."

Alternating is not enough — it must match term-for-term. The odd-power alternating series is

sin θ

; the even-power one is

cos θ

. This bracket has odd powers, so it is

sin θ

Why questions

WHY1. Why does substituting

x = i θ

split the

e^{x}

series cleanly into a real and an imaginary series?

Because

i^{n}

cycles through

1, i, - 1, - i

with period 4: even-

n

terms carry

\pm 1

(real) and odd-

n

terms carry

\pm i

(imaginary), so the terms sort themselves by parity of

n

WHY2. Why are we even allowed to feed a complex number

i θ

into the series for

e^{x}

Because

e^{x}

is entire — its Maclaurin series (from Taylor & Maclaurin Series) converges for every complex input, so the series defines

e^{i θ}

and the substitution is legitimate.

WHY3. Why does Euler's formula make multiplication of complex numbers so easy?

In polar form

z = r e^{i θ}

, multiplying gives

r_{1} r_{2} e^{i (θ_{1} + θ_{2})}

: moduli multiply, arguments add. See Complex Numbers — Polar / Modulus-Argument form.

WHY4. Why must

θ

be in radians and not degrees?

The Maclaurin series

sin x = x - \frac{x ^{3}}{3 !} + \dots

equals the sine function only when

x

is in radians (the derivative

\frac{d}{d x} sin x = cos x

holds only in radians). Degrees would insert a hidden factor of

π /180

WHY5. Why is

e^{i θ}

described as "rotation" while

e^{x}

(real

x

) is "stretching"?

Multiplying by

i

is a

90°

turn, so growth in the imaginary direction is always perpendicular to the current position — that perpendicular push traces a circle instead of a line.

WHY6. Why does replacing

θ

- θ

turn

e^{i θ}

into its conjugate?

cos

is even (

cos (- θ) = cos θ

) and

sin

is odd (

sin (- θ) = - sin θ

), so

e^{- i θ} = cos θ - i sin θ

— same real part, flipped imaginary part, which is exactly conjugation.

WHY7. Why do the roots of unity all lie on the unit circle at equal angular spacing?

Each is

e^{2 π ik / n}

, which has modulus 1 (on the circle) and argument

2 π k / n

(evenly spaced by

2 π / n

). See Roots of Unity.

Edge cases

EC1. What is

e^{i \cdot 0}

, and does the formula behave sensibly there?

e^{0} = cos 0 + i sin 0 = 1 + 0 = 1

. Zero rotation leaves the tip at

+ 1

— the natural starting point on the circle.

EC2. What happens to

e^{i θ}

θ \to \infty

? Does it converge?

It does not converge — it keeps circling the unit circle forever with

∣ e^{i θ} ∣ = 1

. There is no limit; it is bounded but perpetually oscillating.

EC3. Is there any real

θ

for which

e^{i θ} = 0

No. Its modulus is always

1

, and

0

has modulus

0

, so

e^{i θ}

can never be zero — the origin is unreachable.

EC4. For which

θ

e^{i θ}

purely imaginary?

When the real part

cos θ = 0

, i.e.

θ = \frac{π}{2} + k π

. Then

e^{i θ} = \pm i

(top or bottom of the circle).

EC5. If we allow a complex exponent

z = a + bi

, does

∣ e^{z} ∣

still equal 1?

No.

e^{a + bi} = e^{a} (cos b + i sin b)

, so

∣ e^{z} ∣ = e^{a}

. Only the imaginary part

b

rotates; the real part

a

scales the modulus.

EC6. What does

e^{i θ}

give when

θ = 2 π

exactly, and how is that different from

θ = 0

Both give

1

— the same point. The difference is only the journey:

θ = 2 π

has travelled one full loop,

θ = 0

has not moved. As arguments of a complex number they are indistinguishable.

EC7. The hyperbolic analogue: is

e^{θ} = cosh θ + sinh θ

the "real cousin" of Euler's formula?

Yes — dropping the

i

gives

e^{θ} = cosh θ + sinh θ

, the sum of the even and odd parts of

e^{θ}

. See Hyperbolic Functions — $\cosh,\sinh$ vs $\cos,\sin$; the

i

is exactly what converts stretching into spinning.

Recall One-line self-test

If someone claims " $e^{i θ}$ is basically $e$ raised to a big number, so it explodes," what single fact refutes them? ::: $∣ e^{i θ} ∣ = 1$ for all real $θ$ — it lives on the unit circle and never explodes.

Connections

Euler's formula — $e^{i\theta} = \cos\theta + i\sin\theta$ (proof via Taylor series)
Complex Numbers — Polar / Modulus-Argument form
De Moivre's Theorem
Roots of Unity
Taylor & Maclaurin Series
Multiplication as Rotation & Scaling
Euler's Identity $e^{i\pi}+1=0$
Hyperbolic Functions — $\cosh,\sinh$ vs $\cos,\sin$