Identities & phase

Trig identities have a reputation as exam-day hazing. They're actually the algebra of waves: rules for shifting, combining, and disguising sinusoids — and the small set worth knowing prepares you directly for signals and Fourier analysis.

Build the intuition

Phase: where the wave is in its cycle

sin(ωt + φ) is the same wave started earlier by φ — phase is timing expressed as angle. Two speakers playing in phase reinforce; half a cycle out of phase, they cancel (noise-cancelling's whole trick). Phase is the third parameter of every sinusoid, easily forgotten and frequently decisive.

A\sin(\omega t + \varphi)

The one identity family to keep

The angle-sum formulas — sin(a+b) = sin a cos b + cos a sin b and its cosine twin — generate most others (double angles, shifts, products). Read sin(ωt + φ) through it and a phase-shifted wave unfolds into a mix of sin and cos. Most identity gymnastics is this family, dressed up.

\sin(a+b) = \sin a \cos b + \cos a \sin b

Any sin + cos mix is one wave

a sin(ωt) + b cos(ωt) always collapses to a single sinusoid: amplitude √(a² + b²), phase tan⁻¹(b/a) — Pythagoras and arctangent, moonlighting. This is why “amplitude and phase” fully describe any same-frequency combination, and it's the fact Fourier analysis leans on when it splits signals into cos and sin parts.

a\sin\omega t + b\cos\omega t = \sqrt{a^2+b^2}\,\sin(\omega t + \varphi)

Beats: when frequencies almost agree

Add two tones at 440 and 442 Hz and the product identities predict a 441 Hz tone whose loudness throbs twice per second — beats. Musicians tune instruments by listening for the throb to slow and vanish. Identities you can hear.

See it move

InteractiveBuild a wave

Amplitude1.5

Frequency1

y = 1.5 sin(1x): swings 1.5 high (amplitude — loudness, brightness, tide height) and repeats every 6.28 units (period — pitch, frequency, season length).

Amplitude and frequency are two of a wave's three dials — this lesson adds the third, phase, and the algebra for combining all of them.

A worked example

Collapse a mix into one wave

Combine 3 sin(ωt) + 4 cos(ωt).
Amplitude: √(3² + 4²) = 5. Phase: tan⁻¹(4/3) ≈ 53°.
Result:
$3\sin\omega t + 4\cos\omega t = 5\sin(\omega t + 53°)$
The 3-4-5 triangle, conducting an orchestra. Two waves were one wave all along.

Out in the world

Why your voice survives the radio

AM and FM broadcasting work by algebraically combining your audio wave with a carrier wave — the product identities describe exactly which frequencies the mix lands on, and receivers use the same identities backwards to recover the voice. Modulation is applied trig identity.

Common confusion, cleared

“There are dozens of identities to memorize.”

There's one family to know (angle-sum) and a strategy: derive the rest on demand or look them up. Professionals remember structure, not the catalog.

“Phase barely matters next to frequency and amplitude.”

Phase decides whether waves reinforce or annihilate — interference, beamforming, and noise cancellation are pure phase engineering. Ignore it and two perfect speakers can produce silence.

Check yourself

PracticeQuick check

Two identical waves, half a cycle out of phase, are added. The result is…

Recap

Phase = position in cycle; it decides reinforcement vs cancellation.
Angle-sum formulas generate the identity zoo; keep that one family.
a sin + b cos = one wave with amplitude √(a²+b²) and a phase — Fourier's enabling fact.

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