Trigonometric Series

Fourier Analysis · Axiom Academy

Understanding how sines and cosines form an orthogonal basis for periodic functions The sine and cosine functions emerge from circular motion. As a point rotates around the unit circle, its vertical position gives sin(θ) and its horizontal position gives cos(θ). These functions oscillate with period 2π, creating the wave patterns we see in the animation above. 2. Orthogonality - The Key Property Two functions are orthogonal if their inner product (integral of their product) equals zero. This is analogous to perpendicular vectors in geometry. The animation shows how positive and negative areas cancel out over the period, resulting in zero total integral. 3. The Trigonometric Series Formula Using the orthogonality property, any periodic function can be expressed as an infinite sum of sines and cosines with different frequencies. a 0 /2 is the average value (DC component) a n are the cosine coefficients n represents the harmonic number (frequency multiplier) 4. Why This Works - Projection onto Basis Functions The coefficients a n and b n are found by projecting the function onto each basis function, just like finding vector components by dot products. This process extracts each frequency component from the original function. The trigonometric series provides a complete framework for analyzing periodic functions by decomposing them into their frequency components. Sine and cosine functions form an orthogonal basis for periodic functions

This is the written version of the interactive lesson above. See the full Fourier Analysis course.