Author: Rishabh Kumar (IIT + ISI | Global Math Mentor)**
Published: October 2025
Category: Complex Numbers | Algebra, Trigonometry & Geometry

🔹 Introduction

What if exponentials, trigonometry, and complex numbers were all part of the same language?

They are.
And the key that connects them is the most elegant equation in mathematics: eiθ=cos⁡θ+isin⁡θ\boxed{e^{i\theta} = \cos\theta + i\sin\theta}eiθ=cosθ+isinθ​

This is Euler’s Formula — one of the most beautiful results ever discovered.

Euler’s identity unites the most fundamental constants of mathematics — e,i,π,1,0e, i, \pi, 1, 0e,i,π,1,0 — in one breathtaking relationship.

🧭 1️⃣ Derivation of Euler’s Formula

Start with the Taylor Series expansions around x=0x = 0x=0: ex=1+x1!+x22!+x33!+x44!+⋯e^x = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdotsex=1+1!x​+2!x2​+3!x3​+4!x4​+⋯ cos⁡x=1−x22!+x44!−x66!+⋯\cos x = 1 – \frac{x^2}{2!} + \frac{x^4}{4!} – \frac{x^6}{6!} + \cdotscosx=1−2!x2​+4!x4​−6!x6​+⋯ sin⁡x=x−x33!+x55!−x77!+⋯\sin x = x – \frac{x^3}{3!} + \frac{x^5}{5!} – \frac{x^7}{7!} + \cdotssinx=x−3!x3​+5!x5​−7!x7​+⋯

Now, replace xxx with iθi\thetaiθ in the first one: eiθ=1+iθ+(iθ)22!+(iθ)33!+(iθ)44!+⋯e^{i\theta} = 1 + i\theta + \frac{(i\theta)^2}{2!} + \frac{(i\theta)^3}{3!} + \frac{(i\theta)^4}{4!} + \cdotseiθ=1+iθ+2!(iθ)2​+3!(iθ)3​+4!(iθ)4​+⋯

Simplify powers of iii: i2=−1, i3=−i, i4=1,…i^2 = -1, \ i^3 = -i, \ i^4 = 1, \dotsi2=−1, i3=−i, i4=1,…

So: eiθ=(1−θ22!+θ44!−⋯ )+i(θ−θ33!+θ55!−⋯ )e^{i\theta} = \left(1 – \frac{\theta^2}{2!} + \frac{\theta^4}{4!} – \cdots\right) + i\left(\theta – \frac{\theta^3}{3!} + \frac{\theta^5}{5!} – \cdots\right)eiθ=(1−2!θ2​+4!θ4​−⋯)+i(θ−3!θ3​+5!θ5​−⋯) eiθ=cos⁡θ+isin⁡θ\boxed{e^{i\theta} = \cos\theta + i\sin\theta}eiθ=cosθ+isinθ​

✅ Proved via power series expansion!

⚡️ 2️⃣ The Exponential (Euler) Form of a Complex Number

If z=r(cos⁡θ+isin⁡θ)z = r(\cos\theta + i\sin\theta)z=r(cosθ+isinθ),
then by Euler’s formula: z=reiθ\boxed{z = re^{i\theta}}z=reiθ​

where

• r=∣z∣=x2+y2r = |z| = \sqrt{x^2 + y^2}r=∣z∣=x2+y2​ is the modulus,
• θ=arg⁡(z)=tan⁡−1 ⁣(yx)\theta = \arg(z) = \tan^{-1}\!\left(\frac{y}{x}\right)θ=arg(z)=tan−1(xy​) is the argument.

🔹 Example 1

z=1+i3z = 1 + i\sqrt{3}z=1+i3​ r=2, θ=π3r = 2, \ \theta = \frac{\pi}{3}r=2, θ=3π​ z=2eiπ/3\boxed{z = 2e^{i\pi/3}}z=2eiπ/3​

🎯 3️⃣ De Moivre’s Theorem from Euler

Using eiθ=cos⁡θ+isin⁡θe^{i\theta} = \cos\theta + i\sin\thetaeiθ=cosθ+isinθ,
for any integer nnn: (cos⁡θ+isin⁡θ)n=einθ=cos⁡nθ+isin⁡nθ(\cos\theta + i\sin\theta)^n = e^{in\theta} = \cos n\theta + i\sin n\theta(cosθ+isinθ)n=einθ=cosnθ+isinnθ

✅ Hence: zn=rneinθ\boxed{z^n = r^n e^{in\theta}}zn=rneinθ​

Powers of complex numbers correspond to scaling the modulus and rotating by multiples of the angle.

🔹 Example 2

z=2ei30°,z4=24ei120°=16ei120°z = 2e^{i30°}, \quad z^4 = 2^4 e^{i120°} = 16e^{i120°}z=2ei30°,z4=24ei120°=16ei120°

In trigonometric form: z4=16(cos⁡120°+isin⁡120°)=−8+83iz^4 = 16(\cos120° + i\sin120°) = -8 + 8\sqrt{3}iz4=16(cos120°+isin120°)=−8+83​i

🧮 4️⃣ Euler’s Identities

Euler’s formula gives rise to Euler’s identities — some of the most famous equations in mathematics.

1️⃣ eiπ+1=0e^{i\pi} + 1 = 0eiπ+1=0 eiπ=cos⁡π+isin⁡π=−1+0i⇒eiπ+1=0e^{i\pi} = \cos\pi + i\sin\pi = -1 + 0i \Rightarrow e^{i\pi} + 1 = 0eiπ=cosπ+isinπ=−1+0i⇒eiπ+1=0

✅ The most beautiful equation in mathematics, uniting
e,i,π,1,and 0e, i, \pi, 1, \text{and } 0e,i,π,1,and 0.

2️⃣ eiθ=cos⁡θ+isin⁡θe^{i\theta} = \cos\theta + i\sin\thetaeiθ=cosθ+isinθ
✅ Converts trigonometric functions into exponential form.

3️⃣ e−iθ=cos⁡θ−isin⁡θe^{-i\theta} = \cos\theta – i\sin\thetae−iθ=cosθ−isinθ
✅ Conjugate symmetry in exponentials.

🔹 From These, We Derive:

cos⁡θ=eiθ+e−iθ2\cos\theta = \frac{e^{i\theta} + e^{-i\theta}}{2}cosθ=2eiθ+e−iθ​ sin⁡θ=eiθ−e−iθ2i\sin\theta = \frac{e^{i\theta} – e^{-i\theta}}{2i}sinθ=2ieiθ−e−iθ​

✅ Trigonometric functions can be expressed in terms of exponentials —
a key bridge to Fourier analysis, signal processing, and wave mechanics.

🌀 5️⃣ Geometric Interpretation

(Illustration: rotation of point on unit circle by θ, corresponding to eiθe^{i\theta}eiθ)

• Multiplying by eiθe^{i\theta}eiθ rotates a complex number anticlockwise by angle θ.
• ∣eiθ∣=1|e^{i\theta}| = 1∣eiθ∣=1, so rotation doesn’t change magnitude.
• eiπ/2=ie^{i\pi/2} = ieiπ/2=i → a rotation of 90°.
• eiπ=−1e^{i\pi} = -1eiπ=−1 → rotation by 180°.

Exponentials of imaginary numbers represent rotations on the Argand plane.

📘 6️⃣ Key Formulas Summary

🔹 Common Mistakes

1. ❌ Mixing degrees and radians (always use radians in eiθe^{i\theta}eiθ).
2. ❌ Forgetting e−iθ=eiθ‾e^{-i\theta} = \overline{e^{i\theta}}e−iθ=eiθ.
3. ❌ Confusing modulus scaling with argument rotation.
4. ❌ Using wrong sign for θ in conjugates.

🌟 Why It Matters

Euler’s formula and exponential form are used in:

• Electrical engineering & signal processing (phasors, waves)
• Quantum mechanics (wave functions eiθe^{i\theta}eiθ)
• Fourier series & transforms
• IB HL & A Level complex geometry problems
• STEP/MAT conceptual questions

Euler’s formula is not just math — it’s the language of nature’s periodicity.

📘 Learn Beyond the Equation

At Math By Rishabh, Euler’s formula isn’t just proven — it’s understood.

In the Mathematics Elevate Mentorship Program, you’ll:
✅ Derive Euler’s formula from first principles,
✅ Apply it to De Moivre’s theorem & roots of unity,
✅ Visualize rotations and transformations on the Argand plane.

🚀 See exponentials as geometry, not just algebra.
👉 Book your personalized mentorship session now at MathByRishabh.com