From trigonometric functions to Fourier
Start from trigonometric functions on the unit circle and watch single sine waves add up to a wide range of periodic signals. The path begins with high-school trigonometry and ends at Fourier series, a university topic.
- Work Unit circle and trigonometric definitions
Define sin and cos on the unit circle: the geometric starting point for the whole path.
- Work Trigonometric identities and angle sums
Angle sums and identities: the algebra behind combining sine and cosine.
- Work Amplitude, period, and phase of a sinusoid
Read the amplitude, period, and phase of a single sine wave.
- Exam Combining sine and cosine
Exam problem: combine a sin x + b cos x into one sine wave, the first real step into superposition.
- Explore Superposition of trigonometric functions and wave interference
Adding two waves becomes interference, from one dimension to two.
- Explore Fourier series
The end point: a well-behaved (for example, piecewise smooth) periodic function is a constant plus infinitely many sine waves; at a jump the series converges to the average of the two one-sided limits.