Trigonometry — interactive book

The complete Trigonometry book in interactive chapters: theory with all demonstrations, figures that you can drag, all exercises with resolution or answers and entrance exams.

Theory organized into definitions, theorems and demonstrations

Interactive figures in the trigonometric cycle and graphs

All exercises, with resolutions and answers

College entrance exams with immediate correction

Start reading

This interactive book covers trigonometry in ten chapters, from arcs and angles to solving any triangles, and ends with 215 entrance exam tests. Each chapter brings together the theory in boxes, the complete demonstrations, interactive figures in the cycle and graphs and the full list of exercises — those solved with the resolution and those proposed with the answer. Some chapters also provide a historical reading.

The chapters appear in the Trigonometry menu: those at the Elementary II level (I, V and IX) in the Elementary II track and the others in the High School track.

Chapter	Subject	Exercises
I. Arcs and angles	radian, trigonometric cycle, congruous arcs	1 a 22
II. Circular functions	sine, cosine, tangent and other functions; charts and periods	23 a 72
III. Fundamental relationships	relationships between roles and identities	73 a 116
IV. Reduction to the 1st quadrant	symmetries, complementary arcs, parity	117 a 125
v. Notable arcs	regular polygons and exact values	126 a 130
VI. Transformations	addition, double arc, half arc, product	131 a 192
VII. Equations	fundamental, classical equations and inverse functions	193 a 270
VIII. Inequalities	inequalities in sine, cosine and tangent	271 a 320
IX. Right triangles	metric relations and trigonometric ratios	321 a 359
X. Any triangles	laws of sines and cosines, areas, resolution	360 a 419
Entrance exams	review of the entire book	215 tests

How to study: read the theory, work with the figures until you understand what they show and only then open the solutions and answers to the exercises. All answers were checked; where the original template had an error, the answer here explains the difference.

Opening reading

Leitura

Cauchy, the rigor of analysis

Augustin-Louis Cauchy (1789–1857) was born in Paris, weeks after the fall of the Bastille. While still young, he caught the attention of Lagrange and Laplace, who followed his studies. He graduated from the Polytechnic School and, as an engineer, worked from 1810 to 1813 on the works at the port of Cherbourg, which Napoleon was preparing for war against England.

Augustin-Louis Cauchy, in 19th century lithograph.Unidentified author · Wikimedia Commons · Public domain

He returned to Paris to dedicate himself to mathematics and became a professor at the Polytechnic. In 1816, after the restoration of the monarchy, he occupied the position of Monk at the Academy of Sciences, removed for political reasons. A fervent Catholic and a convinced monarchist, he refused to swear allegiance to the new king after the revolution of 1830 and left France, accompanying for some time the deposed king Charles X in exile, who gave him the title of baron.

Cauchy published nonstop — his work is only surpassed in volume by Euler’s — and insisted on rigorous demonstrations. Between 1821 and 1829 he wrote three books, starting with Cours d’analyse (1821), which gave calculus the form it has today, with precise definitions of limit, continuity, derived and integral. Similar ideas appeared at almost the same time with Bolzano, the Czech priest reading Chapter I.

He was also one of the creators of the theory of complex variable functions, he wrote in 1812 a long work on determinants, he proved in 1813 Fermat’s polygonal number theorem (every natural number is the sum of at most three triangular numbers, four square numbers, five pentagonal…), studied polyhedra and, with Navier, founded the mathematical theory of elasticity. Many theorems about infinite series bear his name — and they are what allow the sine and cosine to be written as infinite sums, as Fourier would do with any wave.

Sources: Cauchy (MacTutor) · A.-L. Cauchy, Cours d’analyse de l’École royale polytechnique (1821).