Trigonometry · Lesson 5

Waves, identities, and similarity

Sine and cosine are periodic waves with amplitude, frequency, and phase. Identities are compressed geometry. Cosine similarity is a later preview once vectors arrive.

Core ~28 minDeep study ~78 minWhat do amplitude, frequency, and phase each control?How is wave cosine related to (but not the same as) cosine similarity?

Before this: The unit circle

How this idea was born

Once you can follow a point around a circle, a new human need appears: repeating motion through time. Sound, tides, spinning wheels, and AC power all ask how something comes back again and again. Read circle coordinates against time (or against the angle as a horizontal axis) and waves appear. Periodicity is circular motion written as a repeating graph.

Joseph Fourier (1768-1830) argued that complicated periodic behavior can be built from sines and cosines. MacTutor tracks both the bold heat-analysis idea and the long later fight over rigor. Steal the ethic, not the mythology: break a hard oscillation into simpler waves you can analyze. Identities package geometric facts so you stop re-deriving them every time.

This chapter is the wave-and-identity core of the trigonometry track. Inverse trig and triangle laws come next for solving general triangles. Cosine similarity appears only as an optional preview at the end: same word "cosine," but it needs vectors and dot products from linear algebra before it is a working tool.

First principles

\theta

increases by

2\pi

, the point on the unit circle returns to the same place. Therefore

\sin(\theta + 2\pi) = \sin\theta, \quad \cos(\theta + 2\pi) = \cos\theta

Plotting

\sin\theta

against

\theta

produces a wave that repeats forever.

As θ runs, sin θ oscillates between -1 and 1. Same pattern, forever repeating.

That is why trigonometry models cycles: the circle is the geometry of repetition. On the unit circle, cosine is still the $x$ -coordinate and sine the $y$ -coordinate. The wave graph is that coordinate read out as the angle (or time) runs.

An identity is an equality true for all allowed inputs. The parent identity remains

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

Others (angle-addition formulas, double-angle formulas) expand from geometry or, later, from complex exponentials. On day one, learn what an identity is: reusable structure, not a wall to memorize.

Amplitude, frequency, and phase

A practical signal model is

y(t) = A\sin(\omega t + \phi)

Amplitude $A$ : vertical stretch. Peak height is $|A|$ (from the midline). On the unit circle, plain $\sin\theta$ has amplitude 1; $A\sin\theta$ scales that shadow.
Angular frequency $\omega$ : how fast the angle advances. Period $T = 2\pi/\omega$ (when $\omega > 0$ ). Ordinary frequency $f$ in hertz satisfies $\omega = 2\pi f$ when $t$ is in seconds, so $f = 1/T$ .
Phase $\phi$ : horizontal shift of the wave. Positive $\phi$ (in the $+\phi$ convention above) advances the oscillation; equivalently, $y = A\sin(\omega(t - t_0))$ delays by time $t_0$ when $\phi = -\omega t_0$ .

Period alone (without

\omega

) is the special case of the unit-circle angle: smallest positive

T

with

f(\theta + T) = f(\theta)

, namely

2\pi

for

\sin

and

\cos

Transform checklist for

A\sin(\omega t + \phi)

Start from $\sin t$ (amplitude 1, period $2\pi$ , phase 0).
Replace $t$ by $\omega t$ to compress/stretch time (frequency).
Add $\phi$ inside to slide the graph (phase).
Multiply by $A$ to stretch vertically (amplitude).

Worked intuition

Repetition is circular motion read out

Watch a Ferris wheel rider's height versus time: a sine wave appears. The circle was always there; the wave is the shadow of rotation on one axis. Sirens, tides, and alternating current are circle-shadows in different costumes.

If the wheel turns at constant speed, the angle

\theta

grows linearly with time

t

, so height looks like

\sin(\omega t)

for some angular frequency

\omega

. Frequency and period are reciprocals in the right units: if period is

T

, then

\omega = 2\pi/T

Example - sound. A pure musical tone is often modeled as

A\sin(2\pi f t)

where

f

is frequency in Hz. Middle C is about

261.6

Hz. The sine is not the whole story of timbre (real instruments add overtones), but the skeleton is trig.

Example - tides. Coastal tides are approximately periodic with period near 12.4 hours (lunar semidiurnal). Models superpose several sinusoids with different periods. Fourier's ethic in wet shoes.

Fourier's decomposition ethic

Joseph Fourier (1768-1830) argued that complicated periodic behavior can be built from sines and cosines. MacTutor notes the long later fight over rigor. The ethic worth stealing is bold decomposition: break a hard oscillation into simpler waves you can analyze. Controversies about foundations came later; do not flatten them into "Fourier was wrong" or "Fourier settled everything."

Fourier studied heat diffusion; the mathematical tool outgrew the original problem. That pattern (a technique born for one job, adopted everywhere) repeats across applied math.

Identity as compressed geometry

Double-angle (preview):

\sin(2\theta) = 2\sin\theta\cos\theta

. Geometrically, this encodes how sine behaves when you double the turn. You can derive it from the unit circle with area arguments or accept it as a stored fact until you need the proof.

Angle addition (preview):

\cos(\alpha + \beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta

. The formula looks arbitrary until you see it as "rotate by

\alpha

, then by

\beta

" in matrix language. Algebrica's trigonometric identities page groups these by family instead of dumping a flat list.

Pythagorean rearrangements: From

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

, divide by

\cos^2\theta

to get

1 + \tan^2\theta = \sec^2\theta

(where defined). Each rearrangement is the same circle fact in different algebra clothes.

Amplitude / frequency / phase in examples

Sound:

A\sin(2\pi f t)

with

f

in Hz. Middle C is about

261.6

Hz. Louder means larger

|A|

; higher pitch means larger

f

; starting the tone at a different moment is phase.

AC voltage:

V(t) = V_0\sin(\omega t + \phi)

. Phase differences between circuits matter for power. Same frequency, different

\phi

, can cancel or reinforce when summed.

Shift identity:

y = \sin(\theta - \pi/2)

matches

y = \cos\theta

. Cosine is sine with a quarter-period head start - phase as geometry, not a new function family.

Cosine similarity (preview only)

You do not need this to finish the wave chapter. Skim if curious; return after linear algebra.

On the unit circle,

\cos\theta

is already an

x

-coordinate. Much later, when two vectors

\mathbf{u}, \mathbf{v}

exist, one defines the angle

\phi

between them by

\cos\phi = \frac{\mathbf{u}\cdot\mathbf{v}}{\|\mathbf{u}\|\,\|\mathbf{v}\|}

(when lengths are nonzero). That score is called cosine similarity in ML. It needs the dot product and vector length - topics from linear algebra, not from the unit circle alone.

What you can take now: the word cosine still means adjacent/hypotenuse or

x

on the unit circle. The formula above is a preview of "angle between directions in higher dimensions," not homework for this lesson.

Common confusions

Thinking periodicity means "waves are mysterious." Period

2\pi

is the circle closing. Mystery dissolves into geometry.

Memorizing identity lists without a parent picture. Start from

\sin^2 + \cos^2 = 1

and geometric derivations. Lists without structure fade.

Mixing up amplitude, frequency, and phase. Amplitude is height; frequency is how often; phase is where in the cycle you start. Changing one does not automatically change the others.

Mixing up period and frequency. Period is time (or angle) per cycle; frequency is cycles per unit time. They invert:

f = 1/T

when units align.

Treating cosine similarity as required trig homework. It is an optional preview that needs vectors. Unit-circle cosine comes first.

Overclaiming Fourier invented ML metrics. Fourier's wave decomposition and later cosine similarity share trigonometric ancestry. They are not the same historical object.

History and stories

Fourier's work on heat and series of sines and cosines reshaped applied mathematics. The rigor debates that followed are real history, not a gotcha. Bold ideas often arrive before the community agrees on precise hypotheses.

Lagrange and others pushed back on early Fourier claims about which functions could be expanded. The mathematics was eventually made rigorous (think Dirichlet conditions, Lebesgue theory much later). The lesson for learners: a powerful idea can be right in practice long before the proofs settle.

Strogatz's wave chapters make the worldview feel inevitable: once circular motion and its shadows are vivid, stacking waves becomes a natural next ambition. Treat The Joy of X as pedagogy, not as a substitute for MacTutor when you want biographical detail.

Audio synthesizers and signal processing are living descendants of the decomposition ethic. So are many seasonal models and AC analyses.

Real world

Sound and musical synthesis (amplitude, pitch/frequency, phase onset)
Tides and other approximately periodic natural signals
AC power and phase relationships
Spinning signs and mechanical oscillators tracing sine and cosine in time

Even when a signal is not perfectly periodic, sinusoidal building blocks remain a powerful language for approximation.

JPEG and MP3 use frequency-domain ideas (cousins of Fourier decomposition) to compress data. You do not need the full engineering stack to appreciate the ethic: store waves, not raw wiggles.

Seasonal business data (retail spikes, daylight hours) often get modeled with sinusoidal trends plus noise. The fit is approximate; the trig is a deliberate choice of shape family.

EEG brain waves are labeled alpha, beta, delta by frequency bands. Clinicians read spectra; the underlying language is oscillation.

Open when you want the machine-learning connection. Skip freely.

Field notes, extra examples, and glossary live here when you want more than the core path.

Carry this forward

Sine and cosine are periodic because the circle repeats.
$A\sin(\omega t + \phi)$ packages amplitude, frequency, and phase.
Identities package geometric facts for reuse.
Cosine similarity is a linear-algebra preview, not a prerequisite here.
Next in trigonometry: inverse trig and triangle laws for general triangles.

Go deeper when curiosity hits - videos, essays, and primary trails.

referenceMacTutor - Joseph FourierHeat, series of sines and cosines, and the long fight for foundations.
articleAlgebrica - Trigonometric identitiesCore identities with structure, not memorization fog.
articleAlgebrica - Cosine similarityOptional preview: ML angle score once you have vectors.
video3Blue1Brown - But what is a Fourier series?Visual decomposition of waves.
bookSteven Strogatz - The Joy of XSine Qua Non and the wave worldview.