Euler's Formula

Everything in this course has been building to this moment.

You've seen that sine and cosine come from a point on a circle. You've learned that complex numbers are points on a 2D plane. And you've discovered that multiplying complex numbers rotates them.

Euler's formula is the equation that ties all of this together:

e^jθ = cos(θ) + j·sin(θ)

This single equation is the reason DSP works. Let's understand what it actually says.

What It Says

Read it from right to left first: cos(θ) is the real part, sin(θ) is the imaginary part. Together they trace a point on the unit circle in the complex plane.

Now read it from left to right: e^jθ is a complex exponential. As θ increases, it moves around the unit circle.

Euler's formula says these are the same thing. The complex exponential e^jθ is a point on the unit circle at angle θ.

The Famous Special Case

Set θ = π:

e^jπ = cos(π) + j·sin(π) = −1 + 0 = −1

Rearranging:

e^jπ + 1 = 0

This is Euler's identity, often called the most beautiful equation in mathematics. It connects five fundamental constants (e, j, π, 1, 0) in one equation. But for DSP, the general formula e^jθ = cosθ + j·sinθ is what you'll use every day.

Why It Changes Everything

Before Euler's formula, working with sinusoids meant juggling trig identities. Remember all those formulas from the last lesson? Watch them disappear:

Rotation becomes multiplication. Trig identities become exponent rules. This is why every modern DSP textbook writes signals as complex exponentials, not as sines and cosines.

Extracting Sine and Cosine

By combining e^jθ and its conjugate e^−jθ, you can isolate sine and cosine:

cos(θ) = ¹⁄₂(e^jθ + e^−jθ)

sin(θ) = ¹⁄_2j(e^jθ − e^−jθ)

These are called the inverse Euler formulas. They say that cosine is the average of two spinning arrows (one going counterclockwise, one clockwise), and sine is their difference. You'll see this picture constantly in Fourier analysis. It's why every real sinusoid shows up as two peaks in the frequency spectrum (positive and negative frequency).

Deriving Trig Identities (The Easy Way)

Remember the promise from the last lesson? Here's how you can derive the cosine addition formula using nothing but exponent rules:

Both sum formulas fell out in one step. No memorization needed, just Euler's formula plus FOIL.

The DSP Connection

Here's why this matters so much for signal processing:

• Signals become exponentials: Instead of A·sin(ωt + φ), we write A·e^{j(ωt + φ)}. Simpler algebra.
• The Fourier transform multiplies by e^−jωt: It asks "how much does my signal correlate with a spinning arrow at frequency ω?"
• Filters multiply in the frequency domain: H(ω) is a complex number at each frequency, and its magnitude scales the signal, its angle shifts the phase. That's it.
• Phasors: In circuit analysis, a steady-state sinusoid at frequency ω is fully described by its complex amplitude A·e^jφ. This turns differential equations into algebra.

The next lesson on Complex Exponentials will put this into motion (literally) by exploring what happens when θ = ωt and the point keeps spinning.