0.1 Trigonometry

Trigonometry begins as the study of triangles and becomes the study of rotation and periodicity. The second reading is the one the bookshelf needs. Every oscillation, every wave, every Fourier component is a sine or a cosine, and the few facts that make those manageable — that they are coordinates of a point going around a circle, that they satisfy one quadratic identity, and that sums of angles expand into a fixed pattern — are the entire working content of the subject. This lesson develops them from the unit circle, in the form the later chapters use.

The radian

An angle can be measured in degrees, an arbitrary division of the circle into 360 parts, or in radians, which measure the angle by the arc length it subtends on a circle of radius 1. A full turn is the full circumference $2\pi$ , so $360° = 2\pi$ radians, a right angle is $\pi/2$ , and one radian is about $57.3°$ .

The radian is not a matter of taste. It is the measure that makes the calculus of trigonometric functions clean:

\frac{d}{d\theta}\sin\theta = \cos\theta

holds only when $\theta$ is in radians. In degrees an unwanted factor of $\pi/180$ appears at every differentiation and compounds. Every formula in this project that touches a derivative of a sinusoid — which is most of them — silently assumes radians.

▶ Why the derivative of sine is cosine only in radians Derivation

The derivative rests on the small-angle limit $\lim_{\theta\to 0}\dfrac{\sin\theta}{\theta} = 1$ . Geometrically, for a small angle $\theta$ on the unit circle, the arc length is $\theta$ (the definition of the radian), the chord is $2\sin(\theta/2)$ , and the vertical projection is $\sin\theta$ ; as $\theta\to 0$ the arc and its vertical projection become indistinguishable, so $\sin\theta \to \theta$ . Hence the ratio tends to $1$ .

From the limit, with the angle-sum formula derived below,

\frac{d}{d\theta}\sin\theta = \lim_{h\to 0}\frac{\sin(\theta+h)-\sin\theta}{h} = \lim_{h\to 0}\frac{\sin\theta\cos h + \cos\theta\sin h - \sin\theta}{h}.

Group the terms:

= \sin\theta\,\underbrace{\lim_{h\to 0}\frac{\cos h - 1}{h}}_{0} \;+\; \cos\theta\,\underbrace{\lim_{h\to 0}\frac{\sin h}{h}}_{1} = \cos\theta.

Had $\theta$ been in degrees, the arc subtended by angle $\theta$ would be $\tfrac{\pi}{180}\theta$ , the small-angle limit would read $\sin\theta \to \tfrac{\pi}{180}\theta$ , and that factor would survive into the derivative. Radians are the unit in which the geometric arc is the angle. ✓

The unit circle defines sine and cosine

Place a point $P$ on the circle of radius 1 centred at the origin, at angle $\theta$ measured counter-clockwise from the positive $x$ -axis. Its coordinates are the cosine and sine:

\boxed{\;P = (\cos\theta,\ \sin\theta).\;}

This is the definition to carry. The right-triangle picture — “opposite over hypotenuse” — is the special case in the first quadrant; the circle extends it to all angles, positive and negative, beyond a full turn, and makes the periodicity $\cos(\theta + 2\pi) = \cos\theta$ a triviality, since adding $2\pi$ returns $P$ to where it started.

θ = 0.90 rad (52°)

cos θ = +0.62 sin θ = +0.78 tan θ = +1.26 cos²θ + sin²θ = 1.00

As $\theta$ sweeps, the horizontal coordinate of $P$ traces the cosine curve in the upper panel and the vertical coordinate traces the sine in the lower one. Two readings are worth taking deliberately. First, the sine and cosine are the same curve shifted by a quarter turn: $\cos\theta = \sin(\theta + \pi/2)$ , visible as the quarter-period lead of one trace over the other. Second, at the snap angles $\pi/6, \pi/4, \pi/3$ the coordinates take the values worth committing to memory — $\tfrac12, \tfrac{\sqrt2}{2}, \tfrac{\sqrt3}{2}$ and their partners — because these are the angles that recur in worked problems.

The Pythagorean identity

The single most-used trigonometric fact is that $P$ lies on the unit circle, which is exactly the statement

\boxed{\;\cos^2\theta + \sin^2\theta = 1.\;}

▶ The identity is the equation of the circle Derivation

The circle of radius 1 is the set of points at distance 1 from the origin. For the point $P = (\cos\theta, \sin\theta)$ , the distance to the origin is $\sqrt{\cos^2\theta + \sin^2\theta}$ by the distance formula. Setting that distance to 1 and squaring gives $\cos^2\theta + \sin^2\theta = 1$ for every $\theta$ . There is nothing to compute: the identity is the Pythagorean theorem applied to the right triangle with legs $\cos\theta$ and $\sin\theta$ and hypotenuse 1. ✓

Dividing through by $\cos^2\theta$ or $\sin^2\theta$ gives the two companion identities $1 + \tan^2\theta = \sec^2\theta$ and $\cot^2\theta + 1 = \csc^2\theta$ .

Tangent and the reciprocal functions

The tangent is the slope of the radius to $P$ ,

\tan\theta = \frac{\sin\theta}{\cos\theta},

undefined where $\cos\theta = 0$ — at $\theta = \pi/2, 3\pi/2, \dots$ — where the radius is vertical and the slope diverges. The reciprocal functions $\sec\theta = 1/\cos\theta$ , $\csc\theta = 1/\sin\theta$ , and $\cot\theta = 1/\tan\theta$ are names for combinations that recur; they carry no new content beyond sine and cosine.

Amplitude, period, phase

A pure oscillation is written

x(t) = A\cos(\omega t + \varphi),

and the three constants are read directly off the unit-circle picture with $\theta = \omega t + \varphi$ :

Amplitude $A$ scales the circle from radius 1 to radius $A$ ; it is the peak excursion.
Angular frequency $\omega$ sets how fast $\theta$ winds: the angle advances by $\omega$ radians per unit time, so the period is $T = 2\pi/\omega$ and the ordinary frequency is $f = \omega/2\pi$ .
Phase $\varphi$ is the starting angle at $t = 0$ — a head start around the circle, equivalently a shift of the waveform left by $\varphi/\omega$ in time.

This is the form in which oscillations enter the oscillator chapter of What is sound? and the phasor machinery of Foundations Ch 3. Reading $A$ , $\omega$ , and $\varphi$ off a sinusoid by inspection is a reflex worth having.

The identities that recur

Two derivations downstream depend on the angle-sum formulas

\cos(\alpha+\beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta,\qquad \sin(\alpha+\beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta.

Everything else in the standard table follows from these two. Setting $\beta = \alpha$ gives the double-angle formulas $\cos 2\alpha = \cos^2\alpha - \sin^2\alpha = 1 - 2\sin^2\alpha$ and $\sin 2\alpha = 2\sin\alpha\cos\alpha$ ; rearranging the first gives the half-angle / power-reduction identities $\sin^2\alpha = \tfrac12(1-\cos 2\alpha)$ and $\cos^2\alpha = \tfrac12(1+\cos 2\alpha)$ that turn every squared sinusoid — every energy and intensity integral — into something a $\int$ can dispatch.

▶ Angle-sum formulas from a rotation Derivation

Rotating the plane by angle $\beta$ sends the basis vector $(\cos\alpha,\sin\alpha)$ , which sits at angle $\alpha$ , to the vector at angle $\alpha+\beta$ . The rotation by $\beta$ is the matrix

R_\beta = \begin{pmatrix} \cos\beta & -\sin\beta \\ \sin\beta & \cos\beta \end{pmatrix},

\begin{pmatrix} \cos(\alpha+\beta) \\ \sin(\alpha+\beta) \end{pmatrix} = R_\beta \begin{pmatrix} \cos\alpha \\ \sin\alpha \end{pmatrix} = \begin{pmatrix} \cos\alpha\cos\beta - \sin\alpha\sin\beta \\ \sin\alpha\cos\beta + \cos\alpha\sin\beta \end{pmatrix}.

Reading off the two components gives both formulas at once. The same result drops out in one line from Euler’s formula by equating real and imaginary parts of $e^{i(\alpha+\beta)} = e^{i\alpha}e^{i\beta}$ — the complex-exponential route (refresher: Euler’s formula →) is the reason these identities never need to be memorised once the exponential is in hand. ✓

A closely related pair, the product-to-sum identities, turns products of sinusoids into sums:

\cos\alpha\cos\beta = \tfrac12\bigl[\cos(\alpha-\beta) + \cos(\alpha+\beta)\bigr],

obtained by adding the expansions of $\cos(\alpha\pm\beta)$ . This is the identity behind beats — two nearby tones multiply into a slow envelope times a fast carrier — and behind the orthogonality of sinusoids that makes Fourier series work.

Check yourself

Show that $\cos\theta = \sin(\theta + \pi/2)$ from the angle-sum formula, and say what it means on the unit circle.

Reveal answer

Expand $\sin(\theta + \pi/2) = \sin\theta\cos(\pi/2) + \cos\theta\sin(\pi/2) = \sin\theta\cdot 0 + \cos\theta\cdot 1 = \cos\theta$ . On the circle, advancing the angle by a quarter turn $\pi/2$ sends the vertical coordinate (sine) to where the horizontal coordinate (cosine) was — the two curves are one curve shifted by a quarter period.

⏳ The history — Chords before sines: Hipparchus to the analytic turn

Trigonometry began as a table-making craft for astronomy. Hipparchus of Nicaea (2nd century BCE) is credited with the first table of chords — for each central angle, the length of the chord it cuts on a fixed circle — which Ptolemy systematised in the Almagest (2nd century CE). The chord is a near-relative of the sine: the chord of angle $\theta$ on a unit-diameter circle is $\sin(\theta/2)$ doubled. The half-chord — the modern sine — was the Indian refinement; the Sanskrit jyā (“bowstring”) was transliterated into Arabic and then, by a copyist’s reading of the consonants, mistranslated into Latin as sinus (“fold, bay”), which is the word we still use.

The decisive shift was from geometry to analysis. Leonhard Euler, in the Introductio in analysin infinitorum (Euler 1748), treated sine and cosine as functions of a real variable rather than ratios in a triangle, connected them to the exponential through $e^{i\theta} = \cos\theta + i\sin\theta$ , and in doing so made the entire table of identities consequences of a single algebraic fact. Every identity in this lesson is, after Euler, a corollary of $e^{i\theta}$ .

What we use it for

Trigonometry is the alphabet of the oscillatory bookshelf:

Phasors and complex exponentials repackage $A\cos(\omega t+\varphi)$ as $\operatorname{Re}[A e^{i\varphi} e^{i\omega t}]$ — see Foundations 3.1.
Fourier series and transforms express any signal as a sum of sines and cosines; the product-to-sum identity is what makes the components orthogonal — Foundations Ch 7.
The wave equation’s solutions are sinusoids in space and time, $\cos(\omega t - kx)$ — Sound Ch 4.
Oscillators of every kind, mechanical and acoustic, ring as $A\cos(\omega t + \varphi)$ — Sound Ch 2.
Power-reduction identities turn the squared sinusoids in every energy and intensity calculation into integrable form — Sound Ch 5.

What’s next

The next lesson, 0.2 — Logarithms and exponentials, develops the other half of the pre-calculus toolkit: the exponential function, its inverse the logarithm, and the habit of thinking in decades that the decibel and the octave demand. With both lessons in hand, the single-variable calculus chapter can assume them freely.