Trigonometry

Solving any triangles

Solving any triangle means calculating its main elements: $a$, $b$, $c$, $\hat{A}$, $\hat{B}$ and $\hat{C}$. To do this, it is necessary to provide three information about the triangle, at least one of which is the measurement of a segment linked to it (side, height, median, etc.). The story The first European treatise dedicated entirely to solving triangles, both plane and spherical, was De triangulis omnimodis (“On triangles of all types”), written around 1464 by the German astronomer Regiomontanus and published only in 1533. He organized the solving cases in a very similar way to what we will do here, and the book was the gateway to trigonometry in Renaissance Europe.

October 24, 2026 Read

Heights, medians, angle bisectors and radii

We will deduce formulas that allow us to calculate the notable segments of a triangle — heights, medians, internal angle bisectors, radii of the inscribed and circumscribed circles — knowing only the measurements of the sides. In all of them, the semiperimeter $p = \frac{a + b + c}{2}$ appears. The story The formula for the area on three sides is named after Heron of Alexandria (1st century), who demonstrated it in the Metric, a manual of practical measurements. The Persian mathematician al-Biruni, however, attributed it to Archimedes, two and a half centuries earlier. Heron’s demonstration is purely geometric, with the circumference inscribed; the one we will do here is algebraic, starting from the law of cosines.

October 24, 2026 Read

Projections, areas and law of tangents

Four theorems complete the trigonometric properties of triangles: one decomposes each side into the projections of the other two, two give the area and the last, the law of tangents, relates sums and differences of sides to sums and differences of angles. The story The law of tangents was published by the Danish Thomas Fincke in 1583, in the same book in which he named the tangent (see chapter VIII), and François Viète also used it at the same time. It became very practical in the era of logarithm tables: as it only involves quotients and products, it allows you to solve a triangle knowing two sides and the angle between them without adding squares, which the law of cosines would require. With logarithms, each multiplication or division became an addition or subtraction of numbers taken from the table.

October 24, 2026 Read

Law of cosines and law of sines

The two laws in this topic are the main tools for working with triangles that are not right angles. Both are demonstrated by reducing the problem to right triangles: plotting a height, in the first, and a diameter, in the second. The story In geometric form, the law of cosines is already found in Euclid’s The Elements (book II, propositions 12 and 13), which compare the square on the opposite side to an obtuse or acute angle with the sum of the squares of the other two — without talking about cosine, of course. In trigonometric form, it appears in Miftah al-hisab (“The Key to Arithmetic”, 1427) of the Persian astronomer Jamshid al-Kashi, and that is why the French call it Al-Kashi’s theorem. The law of sines for flat triangles was enunciated and demonstrated in the 13th century by Nasir al-Din al-Tusi, in his Treatise on the quadrilateral, the first work to treat trigonometry as a discipline independent of astronomy (see chapter III).

October 24, 2026 Read

Exercises from chapter IX — Right triangles

Always draw a figure: mark the right angle, the data and the unknown. Then choose the right triangle in which the unknown and the data appear together. Metric relationships 321 Calculate the elements $a$, $h$, $m$ and $n$ indicated in the figure (categories $b = 4$ and $c = 3$). Ative o JavaScript para ver esta figura interativa. See the resolutionBeing $c = 3$ and $b = 4$: $$a^2 = b^2 + c^2 = 16 + 9 = 25 \Rightarrow a = 5$$$$h = \frac{bc}{a} = \frac{4 \cdot 3}{5} = \frac{12}{5}$$$$m = \frac{c^2}{a} = \frac{9}{5}$$$$n = \frac{b^2}{a} = \frac{16}{5}.$$

October 24, 2026 Read

Solving right-angled triangles

Solving a right-angled triangle means calculating its main elements: the acute angles ($\hat{B}$ and $\hat{C}$) and the sides ($a$, $b$, $c$). To do this, it is necessary to provide two information about the triangle, at least one of which is the measurement of a segment connected to it (side, height, median, angle bisector etc.) — only with angles, we know the shape but not the size. The story Solving right triangles was, for centuries, the daily work of surveyors, navigators and artillerymen. To measure angles on the ground, the Englishman Leonard Digges described an instrument he called theodelitus — the ancestor of the theodolite, which to this day appears on the streets on a tripod. The description appeared in Pantometria (“Universal Measurement”), a book that Leonard left unfinished and that his son, Thomas Digges, completed and published in 1571. With a measured angle and a known distance, the trigonometric tables did the rest.

October 24, 2026 Read

Trigonometric ratios in the right triangle

Let’s prove three properties that relate the measurements of the sides to the measurements of the angles of a right-angled triangle. They link the trigonometry of the cycle, which we have studied so far, to the trigonometry of triangles, where it all began. The story It is said that Tales of Miletus (6th century BC) measured the height of the Great Pyramid of Egypt by comparing its shadow with the shadow of a stick: at the same instant, the Sun forms the same angle with both objects, and the ratio between height and shadow is the same for both. This is exactly the idea of this chapter — once an angle is fixed, the ratio between two sides of the right triangle is fixed — although, for Thales, there were still no names or tables for these ratios. The story came to us from much later authors, such as Diogenes Laertius and Plutarch.

October 24, 2026 Read

Metric elements and relationships

We know that a triangle is rectangle when one of its internal angles is right. In this chapter we will develop a geometric-trigonometric theory that allows us to calculate the measurements of segments and angles of a right-angled triangle based on a minimum number of data. The story The metric relationships of the right triangle are among the oldest in mathematics. Babylonian tablets from around 1800 BC, such as the famous Plimpton 322, list suits of numbers that satisfy $a^2 = b^2 + c^2$, and the YBC 7289 tablet gives an excellent approximation of $\sqrt{2}$, the diagonal of the square on its side 1. The first known proof of the theorem, however, is Greek: it appears as the proposition 47 of book I of Euclid’s The Elements (around 300 BC), and tradition attributes it to the school of Pythagoras.

October 24, 2026 Read

Exercises from chapter VIII — Inequalities

In each inequality, draw the cycle: mark the auxiliary line, highlight the appropriate arc and only then write the intervals, starting from $A$. Points that do not satisfy the strict inequality are open. Sine inequalities 271 Solve the inequality $0 \le \operatorname{sen} x < \dfrac{\sqrt{3}}{2}$. See the resolutionThe image of $x$ must be at the intersection of the cycle with the plane range between the straight lines $r$ ($v = 0$) and $s$ ($v = \frac{\sqrt{3}}{2}$), included $r$ and excluded $s$: Ative o JavaScript para ver esta figura interativa. $$2k\pi \le x < \frac{\pi}{3} + 2k\pi \quad\text{ou}\quad \frac{2\pi}{3} + 2k\pi < x \le \pi + 2k\pi.$$

October 24, 2026 Read

Tangent inequalities

For the tangent, the border is no longer a straight line parallel to an axis, but the straight line $OT$ that passes through the center: it cuts the cycle into two diametrically opposite points, and the solutions always come in pairs of arcs symmetric with respect to $O$ — the reflection of the tangent having a period $\pi$. The story The name tangent comes from the axis we use to measure it: the segment $\overline{AT}$ is on the tangent line to the cycle at $A$. The word, in Latin tangens, was introduced by the Danish mathematician Thomas Fincke in the book Geometria rotundi (1583), together with secans. Before, the Arabs already calculated this magnitude as the “shadow” of a vertical rod illuminated by the Sun, in a study linked to sundials (see chapter II).

October 24, 2026 Read

Cosine inequalities

For the cosine, the auxiliary line is vertical: the solutions for $\cos x > m$ are to the right of it, and those for $\cos x < m$, to the left. The story The interval notation we use in this book, with “turned” brackets to indicate an excluded extreme — $\left]a, b\right[$ — is French tradition; it was disseminated in the 20th century by the group Nicolas Bourbaki, a collective pseudonym of French mathematicians who rewrote the foundations of mathematics in a series of books. In English-speaking countries, notation with parentheses, $(a, b)$, is more common.

October 24, 2026 Read

Sine inequalities

Let $f$ and $g$ be two trigonometric functions of the variable $x$. Solving the inequality $f(x) < g(x)$ is to obtain the set $S$ — the solution set or truth set — of numbers $r$ for which $f(r) < g(r)$ is a true sentence. The story The signals $<$ and $>$ appeared printed in 1631, in the book Artis analyticae praxis, published ten years after the death of the English mathematician Thomas Harriot, who was also an astronomer and navigator. The signs $\le$ and $\ge$ (with a dash under the sign) were introduced in 1734 by the French Pierre Bouguer, mathematician and hydrographer. Before that, inequalities were written out in full — “is greater than”, “is less than”.

October 24, 2026 Read