Trigonometry

Exercises from chapter IV — Reduction to the 1st quadrant

Reduction to the 1st quadrant 117 Reduce to the 1st quadrant: a) $\cos 178^\circ$ b) $\operatorname{sen} 251^\circ$ c) $\operatorname{tg} 290^\circ$ d) $\operatorname{cotg}\dfrac{7\pi}{6}$ e) $\sec 1924^\circ$ f) $\operatorname{cossec}\dfrac{23\pi}{6}$ g) $\operatorname{sen}\left(-\dfrac{7\pi}{6}\right)$ h) $\cos\left(-\dfrac{5\pi}{3}\right)$ i) $\operatorname{tg}\left(-\dfrac{3\pi}{4}\right)$ j) $\operatorname{sen}\dfrac{21\pi}{4}$ k) $\cos\dfrac{31\pi}{6}$ l) $\operatorname{tg}\dfrac{11\pi}{3}$ Ver respostaa) $-\cos 2^\circ$ · b) $-\operatorname{sen} 71^\circ$ · c) $-\operatorname{tg} 70^\circ$ · d) $\operatorname{cotg}\dfrac{\pi}{6}$ · e) $-\sec 56^\circ$ · f) $-\operatorname{cossec}\dfrac{\pi}{6}$ · g) $\operatorname{sen}\dfrac{\pi}{6}$ · h) $\cos\dfrac{\pi}{3}$ · i) $\operatorname{tg}\dfrac{\pi}{4}$ · j) $-\operatorname{sen}\dfrac{\pi}{4}$ · k) $-\cos\dfrac{\pi}{6}$ · l) $-\operatorname{tg}\dfrac{\pi}{3}$

October 23, 2026 Read

Reduction identities and even and odd functions

The story The names even function and odd function come from the powers: $x^2$, $x^4$, $x^6$, … do not change when we change $x$ to $-x$, and $x$, $x^3$, $x^5$, … change signs. The terms appear in a work by the young Leonhard Euler on a curve problem proposed at the time, “reciprocal trajectories”, written around 1727. Euler also realized that the cosine behaves like even powers and the sine like odd powers, which is clear in the series he used: $\cos x = 1 - \frac{x^2}{2} + \frac{x^4}{24} - \cdots$ only has even powers, and $\operatorname{sen} x = x - \frac{x^3}{6} + \frac{x^5}{120} - \cdots$, only odd ones.

October 23, 2026 Read

Complementary arcs and the table of values

With the reductions from the previous topic, it is enough to know the functions in the 1st quadrant. A final symmetry — in angle bisector of the 1st quadrant — reduces the work even further: it is enough to know the arcs from $0$ to $\frac{\pi}{4}$, that is, from $0^\circ$ to $45^\circ$. The story The first trigonometric tables were strings: Hipparchus and Ptolemy tabulated the string of each bow, half a degree by half. The Indians began to tabulate the half chord (the sine), and the Arab astronomers added tangents and cotangents, linked to the length of the shadows on sundials. In the 15th century, Regiomontanus calculated sines with 7 digits, and in 1596 the posthumous work of Rheticus, completed by Valentin Otho, brought the six functions of 10 in 10 seconds of arc, with up to 10 numerals — a decades-long work that calculators made unnecessary, but which underpinned navigation and astronomy for centuries. The symmetry of this topic was the first trick of those who assembled a board: calculating the sine up to $45^\circ$, the rest came for free.

October 23, 2026 Read

Reduction of the 2nd, 3rd and 4th quadrants to the 1st

Let’s deduce formulas to calculate the trigonometric functions of $x$, with $x$ outside the 1st quadrant, relating $x$ with some number in the 1st quadrant. The goal is to know $\operatorname{sen} x$, $\cos x$ and $\operatorname{tg} x$ from a table that gives the circular functions of the reals between $0$ and $\frac{\pi}{2}$. Drag $P$ across all quadrants. The point $P'$ of the 1st quadrant is the symmetric point of $P$; Compare the sine and cosine values on the panel. Ative o JavaScript para ver esta figura interativa.

October 23, 2026 Read

Exercises from chapter III — Fundamental relationships

In calculation exercises, the fundamental relationship gives the module and the quadrant gives the sign. In the identities, indicate which of the three methods you used. Calculation of functions from one of them 73 Knowing that $\operatorname{sen} x = \dfrac{4}{5}$ and $\dfrac{\pi}{2} < x < \pi$, calculate the other circular functions of $x$. See the resolutionAs $\dfrac{\pi}{2} < x < \pi \Rightarrow \cos x < 0$, we have: $$\cos x = -\sqrt{1 - \operatorname{sen}^2 x} = -\sqrt{1 - \frac{16}{25}} = -\sqrt{\frac{9}{25}} = -\frac{3}{5}$$$$\operatorname{tg} x = \frac{\operatorname{sen} x}{\cos x} = \frac{\frac{4}{5}}{-\frac{3}{5}} = -\frac{4}{3} \qquad \operatorname{cotg} x = \frac{\cos x}{\operatorname{sen} x} = \frac{-\frac{3}{5}}{\frac{4}{5}} = -\frac{3}{4}$$$$\sec x = \frac{1}{\cos x} = \frac{1}{-\frac{3}{5}} = -\frac{5}{3} \qquad \operatorname{cossec} x = \frac{1}{\operatorname{sen} x} = \frac{1}{\frac{4}{5}} = \frac{5}{4}.$$ 74 Knowing that $\operatorname{cossec} x = -\dfrac{25}{24}$ and $\pi < x < \dfrac{3\pi}{2}$, calculate the other circular functions of $x$. Ver resposta$\operatorname{sen} x = -\dfrac{24}{25}$, $\cos x = -\dfrac{7}{25}$, $\operatorname{tg} x = \dfrac{24}{7}$, $\operatorname{cotg} x = \dfrac{7}{24}$, $\sec x = -\dfrac{25}{7}$

October 23, 2026 Read

Identities and their demonstration

An identity is an equality that holds for all values where both sides make sense. Demonstrating identities is the great training in manipulating trigonometry: each step uses an already proven relationship. The story To demonstrate identities it is necessary to write the functions in short form. Until the 17th century this did not exist: each relationship was said in words — “the sine of the complement of the arc”, “the tangent of the bent arc” — and a one-line identity occupied a paragraph. The abbreviations appeared little by little: Gunter used sin in a figure in 1624, the Frenchman Pierre Hérigone printed it in a book in 1634, and the Englishman William Oughtred wrote something like s co arc for cosine. It was Euler, in Introductio (1748), who established the use of sin., cos., tang. as quantities that are added, multiplied and squared — and it is this use that allows you to manipulate an identity as an algebraic expression.

October 23, 2026 Read

Fundamental relationships

All relations in this topic come from two plane geometry tools: the Pythagorean theorem and the triangle similarity. The figure below shows the three triangles we will use. Drag $P$ and choose the relationship. The panel numerically checks each equality. Ative o JavaScript para ver esta figura interativa. The story In Almagest, Ptolemy already used the equivalent of $\operatorname{sen}^2 x + \cos^2 x = 1$: if he knew the chord of a bow, he obtained the additional bowstring through Pythagoras’ theorem in the triangle inscribed in the semicircle. The relations with tangent and secant appear in the works of Abu al-Wafa (10th century) and Nasir al-Din al-Tusi (13th century), whose Treatise on the quadrilateral made trigonometry a discipline independent of astronomy.

October 22, 2026 Read

Reading: Fourier and heat conduction

Further reading Leitura Fourier and heat conduction Jean-Baptiste Joseph Fourier (1768–1830) was born in Auxerre, France, the son of a tailor. Orphaned before 10 years of age, he was educated at the city’s military school, run by Benedictine monks. As a boy he wrote sermons that priests from other cities used, and at the 13 age he discovered mathematics, in which he quickly excelled. He prepared to enter the Benedictine order, but the Revolution of 1789 changed his plans. Joseph Fourier, in an engraving by Julien-Léopold Boilly.Julien-Léopold Boilly · Wikimedia Commons · Public domain Fourier enthusiastically joined the revolutionary cause—and was nearly guillotined in the Terror of 1794. With the creation of the Escola Normal and the Escola Politécnica in Paris, he became a professor and began the work that would make him famous; from this time is his method for separating the real roots of algebraic equations, a problem that Newton had initiated. In 1798 he accompanied Napoleon on the expedition to Egypt, where he was secretary of the Cairo Institute, directed engineering work and became an expert in Egyptology. Back in France, he was appointed prefect of the Isère department, in Grenoble, and there, between 1804 and 1807, he studied the heat propagation in a solid body. To solve the equations he found, he wrote temperature as an infinite sum of sines and cosines — a type of series development different from Taylor’s, because it uses periodic functions instead of powers. The idea that any functions, even those with “corners”, could be written in this way seemed absurd to Lagrange and other members of the Academy, who received the 1807 memoir with reservations. Fourier insisted, and in 1822 published the Théorie analytique de la chaleur, a landmark in mathematical physics. To the Fourier series They ended up demanding a complete revision of the concepts of function, convergence and integral — work to which Dirichlet, Riemann and Cantor dedicated themselves — and today they are everywhere: in the compression of music and images, in the analysis of earthquakes and in magnetic resonance. Fourier died in Paris in 1830.

October 22, 2026 Read

Exercises from chapter II — Circular functions

Response graphs show a complete period; the dashed curve, when it appears, is the “reference” function ($\operatorname{sen} x$ or $\cos x$). To check your graph, also use $a + b\operatorname{sen}(cx + d)$ explorer. Sine function In exercises 23 to 42, determine the period and the image and plot the graph of a complete period of the given function $f\colon \mathbb{R} \to \mathbb{R}$. 23 $f(x) = -\operatorname{sen} x$. See the resolutionWe build a table in three steps: 1st) we assign values to $x$; 2nd) we associate the value of $\operatorname{sen} x$ to each $x$; 3rd) we multiply $\operatorname{sen} x$ by $-1$. $x$ $0$ $\frac{\pi}{2}$ $\pi$ $\frac{3\pi}{2}$ $2\pi$ $\operatorname{sen} x$ $0$ $1$ $0$ $-1$ $0$ $y$ $0$ $-1$ $0$ $1$ $0$ With this table we obtain 5 points of the graph, which is the symmetric of the sinusoid in relation to the $x$ axis. It is immediate that $\operatorname{Im}(f) = [-1, 1]$ and $p(f) = 2\pi$. Ative o JavaScript para ver esta figura interativa.

October 22, 2026 Read

Cotangent, secant and cosecant

The last three circular functions complete the picture. The cotangent is read on the horizontal axis that touches the cycle at $B$; the secant and cosecant appear when we draw the tangent line to the cycle in $P$ and see where it cuts the axes. The story The names “secant” and “tangent” come from the geometry of the circle: the tangent line touches the circumference (tangere, touch) and the secant line cut (secare, cut). In the definition figure, the segment $\overline{OS}$ lies on a straight line that, when extended, is secant to the cycle, and the segment $\overline{AT}$ lies on the tangent line at $A$. These segments were already tabulated by Arab astronomers: Abu al-Wafa, in the 10th century, organized the six functions and showed how each one derives from the others.

October 22, 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

Cosine function

If the sine is the height of the rotating point, the cosine is its horizontal distance from the center. The two go together: when one is at maximum, the other goes through zero. The story For centuries, the cosine had no name of its own: it was just “the sine of the complement”. Viète called it sinus residuae, and each author wrote differently. In 1620, the Englishman Edmund Gunter proposed, in his Canon triangulorum, the abbreviated form co-sinus — and with it also co-tangens. Four years later, Gunter was the first to use the abbreviation sin in a figure; The names stuck, and the prefix co- became the brand of all “complement” functions.

October 22, 2026 Read