Tangent

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

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

Tangent function

The tangent measures the slope of the straight line that connects the center of the cycle to the point $P$. It is read on a vertical axis that touches the cycle at $A$ — the tangent axis — and, unlike sine and cosine, it can assume any real value. The story The tangent was born far from the circle: it was born from shadows. A vertical stick stuck into the ground, the gnomon, casts a shadow whose length depends on the height of the Sun — and the ratio between the shadow and the stick is, in today’s language, a tangent (or a cotangent, depending on the position of the stick). Because of sundials and prayer times, astronomers in the Islamic world needed these ratios, and the first known shadow tables were made in Baghdad around 860. They are usually attributed to Habash al-Hasib. A century and a half later, al-Biruni brought together everything that was known in the Exhaustive Treatise on Shadows. The name “tangent” would only appear in 1583, with Thomas Fincke (see chapter VIII).

October 22, 2026 Read

Chapter exercises 10 — Circumference and circle

Exercises and resolutions 336 Determine the radius of the center circle $O$, given $AB = 3x - 3$ and $OA = x + 3$. Ver resposta$r = 12$ 337 The circle has a radius of 16 cm and the point $P$ is 7 cm from the center. Determine the distance between $P$ and the circumference. Ver resposta9 cm

October 5, 2026 Read

Tangent segments — circumscribable quadrilaterals

From a point outside a circle two tangents come out, and the two segments to the points of contact have the same length. Who discovered The equality of tangent segments is a direct consequence of proposition III.17 of The Elements of Euclid, which teaches how to draw the tangent through an exterior point. This gives rise to Pitot’s theorem: in a quadrilateral that has an inscribed circle, the sums of the opposite sides are equal. The French engineer Henri Pitot published it in 1725 (see the curiosity about him below). The reciprocal — if the sums of the opposite sides are equal, the quadrilateral has an inscribed circumference — was proven in the 19th century and is usually attributed to Jakob Steiner.

October 5, 2026 Read

Relative positions of two circles

Two circles can be secant, tangent, one inside the other or far apart. The distance between the centers decides which is the case. Who discovered The basic cases are in The Elements of Euclid (c. 300 BC): two distinct circles intersect at a maximum of two points (proposition III.10), and, when they touch, the point of contact is on the straight line that joins the centers (III.11 and III.12). A century later, Apollonius of Perga proposed the most famous problem on the subject: given three circles, draw a fourth tangent to all of them.

October 5, 2026 Read

Relative positions of straight line and circle

A straight line can cut a circle at two points, touch it at just one or pass far. The line that only touches is the tangent. Who discovered The tangent properties are in book III of The Elements by Euclid (c. 300 BC). The proposition III.16 shows that the straight line perpendicular to the radius, at its extremity, touches the circle at a single point, and the III.18 proves the opposite: the tangent is perpendicular to the radius at the point of contact. The word tangent comes from the Latin tangere, “to touch”.

October 4, 2026 Read

Definitions and elements

All points at the same distance from a center: the definition of the circle fits on a line, and yet it hides the number π. Who discovered The circle is one of the oldest forms in human history, from the wheels of Mesopotamia (c. 3500 B.C.) to stone monuments. The rigorous definition is from Euclid (definitions I.15 to I.18 of The Elements): a figure limited by a line whose points are all at the same distance from an interior point, the center. His third postulate guarantees that it is possible to draw a circle with any center and any radius. The first person to accurately calculate the ratio between contour and diameter was Archimedes (3rd century BC).

October 4, 2026 Read