Complex Trigonometry

Why is e^{iπ} + 1 = 0 considered the most beautiful formula in mathematics? How does a radar measure a target's speed from the phase of the reflected signal? How does MP3 compress audio by a factor of 10? All of this is complex trigonometry.

• Digital signal processing: analytic signals for AM/FM demodulation
• Quantum mechanics: wave functions ψ = A·e^{i(kx-ωt)}, the complex exponential
• Electrical engineering: impedances Z = R + iX described through the complex exponential

Предварительные знания

• Law of Cosines and Law of Sines: Arbitrary Triangles
• Trigonometric Substitution and Product-to-Sum Formulas

Trigonometric Functions of a Complex Argument

Boston Dynamics Atlas (2024) uses arctan2 for inverse kinematics: joint angle computation in 1 ms at 1000 Hz control loop. Arctan2 is used in GPS navigation: atan2(dy, dx) gives direction to target within 0.0001° accuracy at 10,000 km range. Sine and cosine can be computed not only for real angles but also for complex numbers z = x + iy. Taylor-series definitions or exponential definitions work for any complex z. The results are no longer bounded by [-1, 1], cosh and sinh emerge naturally.

**Definitions via the complex exponential:** cos(z) = (e^{iz} + e^{-iz}) / 2 sin(z) = (e^{iz} - e^{-iz}) / (2i) **For a purely imaginary argument z = iy:** cos(iy) = (e^{-y} + e^{y}) / 2 = cosh(y) sin(iy) = (e^{-y} - e^{y}) / (2i) = i·sinh(y) **Decomposition formulas for z = x + iy:** cos(x + iy) = cos(x)·cosh(y) - i·sin(x)·sinh(y) sin(x + iy) = sin(x)·cosh(y) + i·cos(x)·sinh(y)

Euler's Formula: e^{iz} = cos z + i·sin z

Euler's formula e^{iθ} = cos θ + i·sin θ is one of the most beautiful in mathematics. It connects the exponential, trigonometry, and complex numbers. At θ = π we get e^{iπ} + 1 = 0, an identity uniting five fundamental constants.

**Derivation via Taylor series:** e^{iz} = 1 + iz + (iz)²/2! + (iz)³/3! + ... = (1 - z²/2! + z⁴/4! - ...) + i(z - z³/3! + z⁵/5! - ...) = cos z + i·sin z **Consequences:** e^{iπ} = -1 → e^{iπ} + 1 = 0 (Euler's identity) cos θ = Re(e^{iθ}), sin θ = Im(e^{iθ}) **Polar form:** z = r·e^{iθ} = r(cos θ + i·sin θ) **Multiplication of complex numbers = rotation + scaling:** z₁·z₂ = r₁·r₂ · e^{i(θ₁+θ₂)}

Hyperbolic Functions and Their Connection via Complex Arguments

Hyperbolic functions cosh and sinh are 'twin siblings' of cosine and sine, connected through imaginary arguments. While trigonometric functions parametrise the unit circle, hyperbolic functions parametrise the unit hyperbola x² - y² = 1.

**Definitions:** cosh x = (e^x + e^{-x}) / 2 (hyperbolic cosine) sinh x = (e^x - e^{-x}) / 2 (hyperbolic sine) tanh x = sinh x / cosh x **Connection with trigonometry:** cos(ix) = cosh(x) sin(ix) = i·sinh(x) cosh(iz) = cos(z) sinh(iz) = i·sin(z) **Identity (analogue of sin²+cos²=1):** cosh²(x) - sinh²(x) = 1 (hyperbolic identity) **Physics:** the catenary (hanging chain) y = a·cosh(x/a)

Applications: Analytic Signals and Instantaneous Phase

An analytic signal is the complex extension of a real signal, enabling computation of instantaneous amplitude and instantaneous phase. It is based on the Hilbert transform and closely tied to Euler's formula. Used in radar systems, ECG analysis, and AM/FM demodulation.

**Analytic signal:** z(t) = x(t) + i·H{x(t)} where H{·} is the Hilbert transform (phase shift of -90°). **Instantaneous amplitude:** A(t) = |z(t)| = √(x² + H{x}²) **Instantaneous phase:** φ(t) = arg(z(t)) = arctan(H{x}/x) **Instantaneous frequency:** ω(t) = dφ/dt **For a tonal signal** x(t) = A·cos(ω₀t + φ₀): H{x} = A·sin(ω₀t + φ₀) z(t) = A·e^{i(ω₀t + φ₀)}, ideal rotation!

Key Ideas

• cos(x+iy) = cos(x)·cosh(y) - i·sin(x)·sinh(y): trigonometry meets hyperbolic functions via imaginary arguments
• Euler's formula: e^{iz} = cos z + i·sin z, consequence, identity e^{iπ}+1=0
• cosh²x - sinh²x = 1: hyperbolas, catenaries, special relativity
• Analytic signal z(t) = x + iH{x}: instantaneous amplitude, phase, and frequency

Related Topics

Complex trigonometry connects analysis, algebra, and physics:

• Trigonometry and 3D Rotations — Quaternions generalise the complex plane to 4 dimensions
• Trigonometric Series and Fourier — Euler's formula is the foundation of the complex Fourier series representation

Вопросы для размышления

• Derive the formula cos(2θ) = cos²θ - sin²θ from De Moivre's theorem (e^{iθ})² = e^{2iθ}.
• Why do hyperbolic functions appear in special relativity? What role does cosh play in Lorentz transformations?
• How are the zeros of sin(z) in the complex plane related to its Taylor series? Can one predict the zeros' locations from the series?

Связанные уроки

• ca-01