Trigonometry

Reading: Euler, analysis incarnate

Further reading Leitura Euler, analysis incarnate Leonhard Euler (1707–1783) was born in Basel, Switzerland. His father, a Protestant pastor, wanted him to pursue a religious career, but the young man was a student of Johann Bernoulli, then the greatest mathematician in Europe, who soon recognized his talent and convinced the family to let him study mathematics. Leonhard Euler, pictured in 1753.Jakob Emanuel Handmann · Wikimedia Commons · Public domain In 1727, at the age of twenty, he went to the newly created Saint Petersburg Academy of Sciences, in Russia, where Daniel and Nicolau Bernoulli, sons of his master, already worked. Around 1738, after an illness, he lost the vision in his right eye. In 1741 he accepted the invitation of Frederick the Great, and spent 25 years at the Berlin Academy; in 1766 he returned to Saint Petersburg, where he stayed until he died. Shortly after his return, a cataract took away his sight in his other eye: Euler spent the last seventeen years of his life practically blind — and it was during this period that he produced almost half of his work, dictating to assistants and making enormous calculations in his head. Euler wrote about almost everything: analysis, number theory, geometry, mechanics, astronomy, optics, music, shipbuilding. His books were so widely read that many of the notations he used have become ours: $e$ for the base of natural logarithms, $\pi$ for the ratio of circumference to diameter, $i$ for $\sqrt{-1}$, $f(x)$ for a function, $\Sigma$ for a sum, and lowercase letters $a$, $b$, $c$ for the sides of a triangle opposite the angles $A$, $B$, $C$. For trigonometry, the decisive work was Introductio in analysin infiniterum (“Introduction to the analysis of infinities”), by 1748. In it, sine and cosine stopped being segments of a circle of arbitrary radius and became functions of a number, measured on a circle of radius 1 — exactly the trigonometric cycle of this book. There also appears the famous formula that links trigonometry to exponential: $$e^{ix} = \cos x + i\operatorname{sen} x.$$Euler died in Saint Petersburg, in 1783, aged 76. The astronomer François Arago said that he “calculated without apparent effort, as men breathe and eagles sustain themselves in the air”; and his contemporaries called him analysis incarnate.

October 24, 2026 Read

Exercises from chapter VII — Equations

For each equation, first ask: what fundamental equation does it reduce to? After writing the families of solutions, check that none of them cancels out a denominator in the original equation. Equation sin α = sin β 193 Solve the equations: a) $\operatorname{sen} x = \operatorname{sen}\dfrac{\pi}{5}$; b) $\operatorname{cossec} x = \operatorname{cossec}\dfrac{2\pi}{3}$; c) $\operatorname{sen} x = 0$; d) $\operatorname{sen} x = \dfrac{1}{2}$; e) $\operatorname{sen} x = -\dfrac{\sqrt{2}}{2}$; f) $\operatorname{sen} x = \dfrac{\sqrt{3}}{2}$; g) $\operatorname{sen} x = 1$; h) $\operatorname{sen} x = -1$. See the resolutiona) $\operatorname{sen} x = \operatorname{sen}\dfrac{\pi}{5} \Rightarrow x = \dfrac{\pi}{5} + 2k\pi$ or $x = \pi - \dfrac{\pi}{5} + 2k\pi$. $\;S = \left\{x \in \mathbb{R} \;\middle|\; x = \dfrac{\pi}{5} + 2k\pi \text{ ou } x = \dfrac{4\pi}{5} + 2k\pi\right\}$. b) $\operatorname{cossec} x = \operatorname{cossec}\dfrac{2\pi}{3} \Rightarrow \dfrac{1}{\operatorname{sen} x} = \dfrac{1}{\operatorname{sen}\frac{2\pi}{3}} \Rightarrow \operatorname{sen} x = \operatorname{sen}\dfrac{2\pi}{3}$, then $x = \dfrac{2\pi}{3} + 2k\pi$ or $x = \pi - \dfrac{2\pi}{3} + 2k\pi$. $\;S = \left\{x \in \mathbb{R} \;\middle|\; x = \dfrac{2\pi}{3} + 2k\pi \text{ ou } x = \dfrac{\pi}{3} + 2k\pi\right\}$. c) $\operatorname{sen} x = 0 = \operatorname{sen} 0 \Rightarrow x = 2k\pi$ or $x = \pi + 2k\pi = (2k + 1)\pi$. $\;S = \{x \in \mathbb{R} \mid x = k\pi\}$. d) $\operatorname{sen} x = \dfrac{1}{2} = \operatorname{sen}\dfrac{\pi}{6}$. $\;S = \left\{x \in \mathbb{R} \;\middle|\; x = \dfrac{\pi}{6} + 2k\pi \text{ ou } x = \dfrac{5\pi}{6} + 2k\pi\right\}$. e) $\operatorname{sen} x = -\dfrac{\sqrt{2}}{2} = \operatorname{sen}\dfrac{5\pi}{4}$, then $x = \dfrac{5\pi}{4} + 2k\pi$ or $x = \pi - \dfrac{5\pi}{4} + 2k\pi$. $\;S = \left\{x \in \mathbb{R} \;\middle|\; x = \dfrac{5\pi}{4} + 2k\pi \text{ ou } x = -\dfrac{\pi}{4} + 2k\pi\right\}$. f) $\operatorname{sen} x = \dfrac{\sqrt{3}}{2} = \operatorname{sen}\dfrac{\pi}{3}$. $\;S = \left\{x \in \mathbb{R} \;\middle|\; x = \dfrac{\pi}{3} + 2k\pi \text{ ou } x = \dfrac{2\pi}{3} + 2k\pi\right\}$. g) $\operatorname{sen} x = 1 = \operatorname{sen}\dfrac{\pi}{2}$ (the two families coincide). $\;S = \left\{x \in \mathbb{R} \;\middle|\; x = \dfrac{\pi}{2} + 2k\pi\right\}$. h) $\operatorname{sen} x = -1 = \operatorname{sen}\dfrac{3\pi}{2}$. $\;S = \left\{x \in \mathbb{R} \;\middle|\; x = \dfrac{3\pi}{2} + 2k\pi\right\}$.

October 24, 2026 Read

Inverse circular functions

The equation $\operatorname{sen} x = \frac{1}{3}$ has infinitely many solutions, but none of them is a notable arc. How to write the solution? We need a name for “the arc whose sine is $\frac{1}{3}$”. The problem is that there are infinitely many arcs like this; so that the name designates a number, we choose an interval where the sine is not repeated. This choice defines the inverse circular functions. The story The notation $\operatorname{arc\,sen} x$ — “the arc whose sine is $x$” — was used by Daniel Bernoulli in 1729 and by the Austrian mathematician Joseph Scherffer in 1772, and gained traction with Lagrange and other authors from the end of the 18th century. The notation $\operatorname{sen}^{-1} x$, common in calculators, was proposed by the English astronomer John Herschel in 1813, by analogy with the inverse of a function; it has the defect of being confused with $\frac{1}{\operatorname{sen} x}$, which is why in this book we use $\operatorname{arc\,sen}$.

October 24, 2026 Read

Solutions on an interval and classical equations

So far we have solved the equations in $\mathbb{R}$, obtaining infinite families of solutions. Often, however, only solutions in a range are of interest — for example, the angles of a triangle, which are between $0$ and $\pi$. And certain equations appear so much that they have their own methods: they are classical equations. The story The equation $a\operatorname{sen} x + b\cos x = c$ is the equation of the sum of two oscillations of the same period: it appears in astronomy (the combination of two circular movements), in wave physics and in electricity (out-of-phase alternating currents). The trick of rewriting $a\operatorname{sen} x + b\cos x$ as a single displaced sinusoid — the auxiliary angle $\theta$ — was already used by 18th century astronomers. It gained industrial importance at the end of the 19th century, when the engineer Charles Steinmetz showed, in 1893, that alternating currents could be added as complex numbers: each sinusoid becomes an arrow with length (the amplitude) and slope (the phase), and adding oscillations becomes adding arrows. This is exactly what we do when writing $a\operatorname{sen} x + b\cos x = \sqrt{a^2 + b^2}\,\operatorname{sen}(x + \theta)$.

October 23, 2026 Read

Fundamental equations

Let $f$ and $g$ be two trigonometric functions of the variable $x$. Solving the equation $f(x) = g(x)$ is determining the set $S$ of numbers $r$ for which $f(r) = g(r)$ is a true sentence. This set is the solution set (or truth set) of the equation. The story Trigonometric equations appeared long before they had that name: Greek, Indian and Arab astronomers needed to find the arc that has a given sine to convert observations into positions in the sky. The most famous of these is the Kepler equation, published in Astronomia nova (1609):

October 23, 2026 Read

Reading: Poncelet, geometry written in prison

Further reading Leitura Poncelet, geometry written in prison Jean-Victor Poncelet (1788–1867) was born in Metz, in eastern France. He studied at the Polytechnic School in Paris, where he was a student of Gaspard Monge, the creator of descriptive geometry, and then at the military engineering school in Metz. In 1812, as an engineering lieutenant, he accompanied Napoleon’s army in the invasion of Russia. Jean-Victor Poncelet.Smithsonian Libraries · Wikimedia Commons · Public domain During his retreat from Moscow, he was presumed dead in the battle of Krasnoi and ended up a prisoner. Taken on foot in winter to Saratov, on the banks of the Volga, he spent around eighteen months in captivity. Without books, he reconstructed from memory what he had learned with Monge and went further: he began to develop projective geometry, the study of the properties of figures that do not change when they are projected from one plane to another — the same properties that Desargues and Pascal had begun to explore in the 17th century. Released in 1814, he returned to France and began publishing his results, first on inscribed polygons circumscribed by conics. The great work, the Traité des propriétés projectives des figures (“Treatise on the projective properties of figures”), came out in 1822. In it, Poncelet showed that properties such as the alignment of points or tangency are preserved through projection, and systematically used the relationship between pole and polar of a conic, the basis of the duality principle: in flat projective geometry, every theorem about points and lines remains true when the words “point” and “line” are exchanged. These ideas paved the way for the study of projective invariants, such as the double ratio (or anharmonic) of four points, developed by Möbius and Chasles, and for the program of Felix Klein, which in 1872 classified geometries by the groups of transformations that preserve them. Poncelet was also an important engineer: he designed hydraulic wheels and taught applied mechanics. Died in Paris, in 1867.

October 23, 2026 Read

Exercises from chapter VI — Transformations

This is the chapter with the most demonstration exercises. Before opening a solution, decide which family of formulas to use: addition, multiplication, division or transformation into a product. Addition formulas 131 Calculate the values of: a) $\cos 15^\circ$; b) $\operatorname{sen} 105^\circ$; c) $\operatorname{tg} 75^\circ$; d) $\sec 285^\circ$. See the resolutiona) $\cos 15^\circ = \cos(45^\circ - 30^\circ) = \cos 45^\circ \cos 30^\circ + \operatorname{sen} 45^\circ \operatorname{sen} 30^\circ = \dfrac{\sqrt{2}}{2} \cdot \dfrac{\sqrt{3}}{2} + \dfrac{\sqrt{2}}{2} \cdot \dfrac{1}{2} = \dfrac{\sqrt{6} + \sqrt{2}}{4}$ b) $\operatorname{sen} 105^\circ = \operatorname{sen}(60^\circ + 45^\circ) = \operatorname{sen} 60^\circ \cos 45^\circ + \operatorname{sen} 45^\circ \cos 60^\circ = \dfrac{\sqrt{3}}{2} \cdot \dfrac{\sqrt{2}}{2} + \dfrac{\sqrt{2}}{2} \cdot \dfrac{1}{2} = \dfrac{\sqrt{6} + \sqrt{2}}{4}$ c) $\operatorname{tg} 75^\circ = \operatorname{tg}(45^\circ + 30^\circ) = \dfrac{\operatorname{tg} 45^\circ + \operatorname{tg} 30^\circ}{1 - \operatorname{tg} 45^\circ \operatorname{tg} 30^\circ} = \dfrac{1 + \frac{\sqrt{3}}{3}}{1 - \frac{\sqrt{3}}{3}} = \dfrac{3 + \sqrt{3}}{3 - \sqrt{3}}$ $\left(= 2 + \sqrt{3}\right)$ d) $\sec 285^\circ = \sec 75^\circ = \dfrac{1}{\cos 75^\circ} = \dfrac{1}{\cos(45^\circ + 30^\circ)} = \dfrac{4}{\sqrt{6} - \sqrt{2}}$ $\left(= \sqrt{6} + \sqrt{2}\right)$

October 23, 2026 Read

Transformation into product

In elementary algebra, transforming a polynomial into a product (factoring) is a resource of enormous practical importance: $x^2 - 2x = x(x - 2)$ highlighting $x^2 - 4 = (x + 2)(x - 2)$ difference of squares $x^2 \pm 4x + 4 = (x \pm 2)^2$ perfect square trinomials $x^3 + 8 = (x + 2)(x^2 - 2x + 4)$ sum of cubes $x^3 - 8 = (x - 2)(x^2 + 2x + 4)$ difference of cubes $x^3 \pm 3x^2 + 3x \pm 1 = (x \pm 1)^3$ perfect cubes We will often apply these features to trigonometry, with transformations like $\operatorname{sen}^2 x - 2\operatorname{sen} x = \operatorname{sen} x(\operatorname{sen} x - 2)$ and $\operatorname{sen}^2 x - \cos^2 x = (\operatorname{sen} x + \cos x)(\operatorname{sen} x - \cos x)$. In addition to algebraic resources, trigonometry has formulas that allow you to complete a factorization — in the example above, also factor $\operatorname{sen} x + \cos x$ and $\operatorname{sen} x - \cos x$.

October 23, 2026 Read

Multiplication and division formulas

By doing $b = a$ in the addition formulas, we obtain the double arc functions; reversing these formulas, those of the arch half. These are the tools that allow, for example, calculating $\operatorname{sen}\frac{\pi}{8}$ from $\cos\frac{\pi}{4}$. The story In 1593, Belgian mathematician Adriaan van Roomen challenged European mathematicians to solve an equation of degree 45. François Viète, lawyer and advisor to the French king Henry IV, realized that it was the expression of $\operatorname{sen} 45\theta$ in function of $\operatorname{sen}\theta$ and solved it with the formulas of multiple arcs, which he himself had systematized. The half bow formulas are even older: Ptolemy used them to pass from the string of $12^\circ$ to those of $6^\circ$, $3^\circ$, $1{,}5^\circ$ and $0{,}75^\circ$ in the construction of his tablet.

October 23, 2026 Read

Addition formulas

Let’s deduce formulas to calculate the trigonometric functions of the sum $(a + b)$ and the difference $(a - b)$ of any two real numbers $a$ and $b$, known as the circular functions of $a$ and $b$. The story In Almagest, Ptolemy proved that, in an inscribed quadrilateral, the product of the diagonals is equal to the sum of the products of the opposite sides — Ptolemy’s theorem. Applied to a quadrilateral with a side equal to the diameter, it gives exactly the chord formulas for the sum and difference of two arcs, which Ptolemy used to assemble his half-by-half degree table. In the form of sines, the formulas appear with the Persian astronomer Abu al-Wafa (10th century), whom we already met in chapter II. The demonstration we will do — based on the distance between two points — uses Descartes’ analytical geometry and is much more recent.

October 23, 2026 Read

Exercises from Chapter V — Notable Arches

126 Calculate $\operatorname{sen} 15^\circ$, $\cos 15^\circ$ and $\operatorname{tg} 15^\circ$. See the resolution$\operatorname{sen} 15^\circ = \operatorname{sen}\dfrac{\pi}{12} = \dfrac{\ell_{12}}{2}$. Let us calculate $\ell_{12}$: let $Q$ be a vertex of the inscribed regular hexagon and $P$ be the midpoint of the arc on one side of the hexagon, so that $PQ = \ell_{12}$. The radius $OP$ cuts the side of the hexagon at its midpoint $R$, perpendicularly. Ative o JavaScript para ver esta figura interativa. In triangle $PQR$, rectangle in $R$: $$PQ = \ell_{12} \qquad QR = \frac{\ell_6}{2} = \frac{1}{2} \qquad RP = OP - OR = 1 - \frac{\sqrt{3}}{2}$$(since $OR$ is the height of equilateral triangle beside 1). Then $$\ell_{12}^2 = \left(\frac{1}{2}\right)^2 + \left(1 - \frac{\sqrt{3}}{2}\right)^2 = \frac{1}{4} + 1 - \sqrt{3} + \frac{3}{4} = 2 - \sqrt{3}$$$$\operatorname{sen}\frac{\pi}{12} = \frac{\ell_{12}}{2} = \frac{\sqrt{2 - \sqrt{3}}}{2} \qquad \cos\frac{\pi}{12} = \sqrt{1 - \frac{2 - \sqrt{3}}{4}} = \frac{\sqrt{2 + \sqrt{3}}}{2} \qquad \operatorname{tg}\frac{\pi}{12} = \frac{\sqrt{2 - \sqrt{3}}}{\sqrt{2 + \sqrt{3}}}.$$Note: in chapter VI we will see that these values are also written $\operatorname{sen} 15^\circ = \dfrac{\sqrt{6} - \sqrt{2}}{4}$, $\cos 15^\circ = \dfrac{\sqrt{6} + \sqrt{2}}{4}$ and $\operatorname{tg} 15^\circ = 2 - \sqrt{3}$.

October 23, 2026 Read

The sine of π/n and notable arcs

We will verify that the circular functions of the reals $x = \dfrac{\pi}{n}$, with $n \in \mathbb{N}$ and $n \ge 3$, can be calculated from $\ell_n$, the side of the regular polygon with $n$ sides inscribed in the cycle. The story In book I of Almagest, Ptolemy constructed his table of chords exactly like this: he started from the sides of inscribed regular polygons — triangle, square, pentagon, hexagon and decagon —, which Euclid’s geometry knew how to calculate, and obtained the other arcs through formulas for sum, difference and half of arcs, which we will see in chapter VI. The chord of an arc $2\alpha$ in a circle of radius $R$ is $2R\operatorname{sen}\alpha$: the theorem in this chapter is the modern translation of this idea.

October 23, 2026 Read