Pre-calculus: trigonometry and logarithms

Radians, the unit circle, and the identities; logs, exponentials, and thinking in decades — the working subset.

The calculus chapters that follow assume two pieces of machinery so thoroughly that they are rarely stated: trigonometry and logarithms. Both are pre-calculus in the curricular sense, but neither is elementary in how often it is used, and both are routinely the place a derivation stalls — not at the calculus, but at a half-remembered identity or a logarithm law. This short chapter restores the two of them to working order. It is a refresher, not a first course: if $A\cos(\omega t + \varphi)$, $\cos^2 + \sin^2 = 1$, and $\log(ab) = \log a + \log b$ are reflexes, skip it.

The selection is deliberate. The rest of the bookshelf runs on oscillations, so trigonometry is the language of every wave, phasor, and Fourier component. And it runs on quantities that span many orders of magnitude — sound pressures from a threshold whisper to a jet engine, frequencies across ten octaves — so it is built on logarithms and the decibel. These are the two pre-calculus tools the other books cannot do without.

• 0.1 Trigonometry — the radian, the unit-circle definition of sine and cosine, the Pythagorean identity, the amplitude–period–phase reading of $A\cos(\omega t + \varphi)$, and the handful of identities (angle-sum, double-angle, product-to-sum) that recur everywhere.
• 0.2 Logarithms and exponentials — exponential and logarithm laws, the natural base $e$ and why it is natural, change of base, exponential growth and decay, and thinking in decades: octaves, log axes, and the decibel.