Text classes and LaTeX formulas

The themes of school trigonometry are organized into 30 classes, with definitions, demonstrations, editable formulas in LaTeX and 60 commented exercises:

Step Classes What to study
Fundamental I 3 Turns, corners, arcs and triangles in preparatory activities
Fundamental II 6 Angles, polygons, notable values, Pythagoras and trigonometric ratios
High School 21 Cycle, functions, identities, transformations, equations, inequalities and any triangles

The conversion map relates the 51 items of the theoretical summary to the classes. These classes are the short didactic version. The interactive Trigonometry book brings the full theory, with all demonstrations, interactive figures, 419 exercises with resolution or answer and 215 entrance exams, organized in the school tracks below.

Study path for children

The following explanations and activities form an original didactic route, organized by stage school. The interactive book has its own summary by chapters.

Choose your step

A mathematics that begins with a turn

How to find out the height of a tree without climbing it? Why can a wheel draw a wave? How to measure the distance to the other bank of a river? Trigonometry relates turns, angles and lengths to answer questions like this.

The existing introductory course brings together ten chapters, commented demonstrations, three figures and 70 exercises with resolution, including a final review. It remains available as complementary reading to the new classes per stage. The activities use paper, pencil, ruler, string and a paper disc. A calculator can help with approximations; deductions from the exact values ​​are made in the text.

Who is this book for?

The entrance was designed for children from approximately 10 years old, accompanied when necessary. This does not mean that all content belongs to this age: equations, identities and general formulas are deepening for the end of Elementary School and High School.

  • Explore: draw, fold, measure and understand concrete examples. It is possible to start without knowing algebra.
  • Investigate: use fractions, proportions, negative numbers, roots and coordinates. Read the demonstrations and do the letter exercises when you are ready.
  • Go deeper: work on transformations, equations, inequalities and special measurements of triangles with the support of a teacher or adult.

Each chapter starts with a concrete situation, explains its tools and presents a demonstration. Do sessions of 20 to 30 minutes and revisit an idea before moving forward. The sequence can take up several weeks; It’s not a race.

Before you start

You will use multiplication, division, fractions and measurements in centimeters or meters. When a letter appears, it will be the name of a measurement or a number that may vary. The symbol $\sqrt9$ represents the positive square root of 9, which is 3. A power like $a^2$ means $a\cdot a$.

In the plane, $(x,y)$ indicates first the horizontal position and then the vertical. The number $\pi$ is approximately $3{,}14159$. We use $\operatorname{sen}$, $\cos$ and $\tan$ for sine, cosine and tangent; on some calculators the buttons are sin, cos and tan. Check whether the device is in degrees (DEG) or radians (RAD) before calculating.

Additional readings: introductory route

The ten themes follow the ten chapters of the interactive book. The texts, examples, figures and exercises below were written for this introductory route, which selects the central results of each chapter; the full version, with all the exercises, is in the book.

Theme Our route Step
Spins and angles 1. The mathematics of turns Fundamental II; radians in high school
Notable triangles and angles 5. Triangles and polygons Fundamental II
Right triangles 9. Measuring without reaching Fundamental II
Circular functions 2. The wheel that makes waves High School
Fundamental relationships 3. The pieces fit together High School
Symmetries 4. A table for the entire lap High School
Transformations 6. Building new formulas High School
Equations 7. Hidden angles High School
Inequalities 8. Pieces of the lap High School
Any triangles 10. The complete toolbox High School
Revision Ten arrival challenges High School

For younger readers, start with the Fundamental I and advance to Fundamental II. The high school track assumes familiarity with Pythagoras; see the demonstration of chapter 9 before fundamental relationships, if necessary.

How to study a demonstration

A proof explains why a statement holds in all the cases described. Copy the figure, name the measurements and redo each passage. Ask: “What definition or outcome enabled this step?” An example helps to understand, but alone does not demonstrate a general rule.

In the exercises, write the reasoning and unit. Then open the resolution and compare the paths. A different answer may be correct if it is equivalent and respects the conditions of the problem.

Small dictionary

Word Meaning
Hypothesis Starting information for an argument
Identity Equality valid across the declared domain
Domain Set of allowed inputs of a function
Period Positive shift that repeats all values ​​of a function
Similarity Same shape, with corresponding proportional sides
Approximation Value close to exact, indicated by $\approx$

For the accompanying adult: first ask for a prediction, then a drawing and only then an account. Treat the error as a clue as to what needs to be revised. The final chapters can wait until the algebraic tools make sense to the reader.

Start with chapter 1 →

Extra activity: measuring without crossing

In the triangulation activity in the field, use a 45° triangle to find a distance without reaching the measured point.

Interactive book

The ten chapters of the interactive Trigonometry book bring the complete theory, with all the demonstrations, figures that you can drag and the complete list of exercises, with resolution or answer. Chapters I, V and IX appear in the Fundamental II track; the others, in high school. In the end, 215 entrance exams review all the content.