Sine Function

📐 Definition

For $x \in \mathbb{R}$ (extended to $x \in \mathbb{C}$ by analytic continuation), the sine function is defined by

\sin x = \frac{e^{ix} - e^{-ix}}{2i}

It represents the imaginary part of the complex exponential and forms a fundamental odd sinusoid.

Domain and Codomain

The sine function is entire on $\mathbb{C}$ . For real $x$ , its range satisfies $-1 \le \sin x \le 1$ ; for complex $x$ , values lie in $\mathbb{C}$ .

⚙️ Key Properties

Periodicity and Parity

\sin(x + 2\pi k) = \sin x, \qquad \sin(-x) = -\sin x, \qquad k \in \mathbb{Z}

The sine function repeats every $2\pi$ and is odd with respect to the origin.

Derivatives and Integrals

\frac{d}{dx}\sin x = \cos x, \qquad \int \sin x\, dx = -\cos x + C

Sine and cosine form a phase-shifted pair under differentiation and integration.

Angle Addition Formula

\sin(x + y) = \sin x \cos y + \cos x \sin y

This identity governs phase shifts, Fourier synthesis, and rotational composition.

Pythagorean Identity with Cosine

\sin^2 x + \cos^2 x = 1

Sine and cosine coordinate points on the unit circle and encode orthogonal modes.

Power Series Expansion

\sin x = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!}

The series converges for all complex $x$ and provides accurate local approximations.

🎯 Special Values

$x$	$\sin x$	Notes
$0$	$0$	$\lim_{x \to 0} \frac{\sin x}{x} = 1$
$\tfrac{\pi}{6}$	$\tfrac{1}{2}$	—
$\tfrac{\pi}{2}$	$1$	maximum on the real line
$\pi$	$0$	zero crossing
$x = n\pi$	$0$ for all $n \in \mathbb{Z}$	infinite zero lattice

The sine function pairs with cosine to span real-valued periodic bases and can be recovered from the complex exponential $e^{ix}$ via

\sin x = \Im\!\left(e^{ix}\right)

Usage in Oakfield

Oakfield uses sine heavily in its stimulus operators and visualization:

Sinusoidal stimulus family (stimulus_sine, stimulus_standing, stimulus_chirp) uses sin/sincos to generate oscillatory forcing with configurable phase, frequency, and wavenumber.
Carrier modulation: Gabor and spectral-line stimuli use sinusoidal carriers under Gaussian envelopes or harmonic sums.
GPU visuals use sine/cosine for phase-to-color mappings and oscillatory effects in shaders.

Historical Foundations

📜 Early Trigonometry

The sine function emerged from earlier chord-based methods in Greek astronomy and was developed in a half-chord form in Indian mathematical astronomy, where tabulated values supported astronomical computation.

🔬 Transmission and Standardization

Sine tables and identities were refined in medieval Islamic mathematics and transmitted to Europe through translation, where the function became standard in navigation, astronomy, and analysis.

🌍 Modern Perspective

In modern analysis, $\sin x$ is treated as an entire function defined by its power series or via the complex exponential, and it serves as a foundational building block in Fourier analysis and wave modeling.

📚 References

NIST Digital Library of Mathematical Functions (DLMF), §4
Abramowitz & Stegun, Handbook of Mathematical Functions