Cosine Function

📐 Definition

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

\cos x = \frac{e^{ix} + e^{-ix}}{2}

It represents the real part of the complex exponential and forms a fundamental even sinusoid.

Domain and Codomain

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

⚙️ Key Properties

Periodicity and Parity

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

Cosine repeats every $2\pi$ and is even about the origin.

Derivatives and Integrals

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

Cosine and sine rotate into each other under differentiation and integration.

Angle Addition Formula

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

This governs phase shifts, wave superposition, and rotational composition.

Phase Shift Relation with Sine

\cos x = \sin\!\left(x + \tfrac{\pi}{2}\right)

Cosine represents a quarter-cycle shift of the sine function.

Pythagorean Identity with Sine

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

Cosine and sine coordinate points on the unit circle and encode orthogonal modes.

Power Series Expansion

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

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

🎯 Special Values

$x$	$\cos x$	Notes
$0$	$1$	local approximation $\cos x \approx 1 - \tfrac{x^2}{2}$ near $0$
$\tfrac{\pi}{3}$	$\tfrac{1}{2}$	—
$\tfrac{\pi}{2}$	$0$	quarter-cycle zero
$\pi$	$-1$	minimum on the real line
$x = \tfrac{\pi}{2} + n\pi$	$0$ for all $n \in \mathbb{Z}$	zeros at odd quarter-cycles

Cosine pairs with sine to form orthogonal periodic bases and can be recovered from the complex exponential $e^{ix}$ via

\cos x = \Re\!\left(e^{ix}\right)

Usage in Oakfield

Oakfield uses cosine alongside sine for efficient oscillatory synthesis:

Stimulus kernels use sincos/cos to build carriers and standing-wave components (e.g. sinusoidal family, spectral lines, random Fourier features).
Complex rotation factors often use (cos θ) + i (sin θ) when applying phase rotations to complex stimuli.
Shader phase view ultimately relies on trig functions when mapping phase angles into color space.

Historical Foundations

📜 Early Trigonometry

Cosine is historically intertwined with sine through complementary angles (cosine as the sine of a complement), arising from tabulated trigonometric values used in astronomy.

🔬 Transmission and Standardization

Through medieval Islamic mathematics and subsequent European adoption, cosine became a standard tool for geometric computation, navigation, and harmonic analysis.

🌍 Modern Perspective

In modern analysis, $\cos x$ is treated as an entire function defined by its power series or via the complex exponential, and it appears ubiquitously in Fourier methods, oscillation theory, and solutions of linear differential equations.

📚 References

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