0.1 Trigonometry

Trigonometry begins as the study of triangles and becomes the study of rotation and periodicity; it is the second reading that the rest of the bookshelf relies on. Every oscillation, every wave, every Fourier component is a sine or a cosine, and three facts about them — that they are the coordinates of a point moving around a circle, that they obey one quadratic identity, and that sums of angles expand in a fixed pattern — carry nearly all the 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, the arc length the angle subtends on a circle of radius 1. A full turn is the whole circumference $2\pi$ , so $360° = 2\pi$ radians, a right angle is $\pi/2$ , and one radian is about $57.3°$ .

Radians are the measure in which the calculus of the trigonometric functions is simplest. The relation

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

holds only when $\theta$ is in radians; measured in degrees, a factor of $\pi/180$ enters at each differentiation and accumulates. Every derivative of a sinusoid in this book is therefore taken with the angle in 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 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).\;}

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

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 in the upper panel and the vertical coordinate the sine in the lower one. Two features stand out. First, 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 $\theta = \pi/6,\ \pi/4,\ \pi/3$ the coordinates take the standard values $\tfrac12$ , $\tfrac{\sqrt2}{2}$ , $\tfrac{\sqrt3}{2}$ and their complements, the values that recur throughout the subject.

The Pythagorean identity

That $P$ lies at distance $1$ from the origin is, written in coordinates, the identity

\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 $P = (\cos\theta, \sin\theta)$ , 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 $\theta$ . 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 three reciprocal functions are

\sec\theta = \frac{1}{\cos\theta},\qquad \csc\theta = \frac{1}{\sin\theta},\qquad \cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta}.

Each is built from sine and cosine and introduces no new quantity, but the names are geometric and the picture is worth keeping. Draw the line tangent to the unit circle at $(1,0)$ — the vertical line $x = 1$ — and extend the radius through $P$ until it meets that line at $(1,\tan\theta)$ . The segment of the tangent line from $(1,0)$ to that point has length $\tan\theta$ , the tangent; the segment from the origin out to the same point runs along a line that cuts the circle — a secant — and has length $\sqrt{1+\tan^2\theta} = \sec\theta$ . The cotangent and cosecant are the identical construction against the horizontal tangent line $y = 1$ . The reciprocal relations fix their range and singularities at a glance: since $\lvert\cos\theta\rvert \le 1$ , the secant has $\lvert\sec\theta\rvert \ge 1$ and diverges wherever $\cos\theta = 0$ ; likewise $\lvert\csc\theta\rvert \ge 1$ , diverging where $\sin\theta = 0$ . The companion identities $1+\tan^2\theta = \sec^2\theta$ and $1+\cot^2\theta = \csc^2\theta$ are the Pythagorean identity rewritten in these variables.

One point of notation deserves emphasis, because it is a common confusion: the cotangent and the arctangent are not the same thing. The cotangent is the reciprocal of the tangent, a number computed from an angle,

\cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta},

whereas the arctangent $\arctan x$ (also written $\tan^{-1} x$ ) is the inverse function of the tangent — the angle whose tangent is $x$ , recovered from a slope. The two satisfy quite different relations,

\tan\theta \cdot \cot\theta = 1, \qquad \text{but} \qquad \tan(\arctan x) = x,

so $\cot\theta \neq \arctan\theta$ in general. The trap is the notation $\tan^{-1}$ , where the superscript $-1$ means functional inverse, never reciprocal — $\tan^{-1} x = \arctan x$ , not $1/\tan x$ . The proper inverse trig functions $\arcsin$ , $\arccos$ , $\arctan$ each undo their function on a restricted range (so the inverse is single-valued), and are what we reach for whenever an equation must be solved for the angle.

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.

The identities that recur

The rest of the standard table follows from 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.

Setting $\beta = \alpha$ in them gives the double-angle formulas $\cos 2\alpha = \cos^2\alpha - \sin^2\alpha = 1 - 2\sin^2\alpha$ and $\sin 2\alpha = 2\sin\alpha\cos\alpha$ ; solving the first for $\sin^2$ or $\cos^2$ gives the half-angle / power-reduction identities $\sin^2\alpha = \tfrac12(1-\cos 2\alpha)$ and $\cos^2\alpha = \tfrac12(1+\cos 2\alpha)$ , which turn a squared sinusoid — every energy and intensity integral — into a form that integrates directly.

▶ 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}.

The two components are the two formulas. They drop out equally 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 →) reduces the whole table of angle identities to the single rule that exponentials add their exponents. ✓

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 identity is behind beats — two nearby tones combine 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 underlies the oscillatory content of the rest of the 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.

Drill

Rote recall of the core identities, as a spaced-repetition deck. Reveal each card, then grade yourself — Again / Hard / Good / Easy — and SM-2 schedules when it returns. Progress is shared with the Foundations study deck.

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.