Trigonometry. Introduction and basic concepts

05/05/2026

The online trigonometry simulations on this page will serve as an introduction to trigonometry. They will help you to better understand the basic concepts and fundamentals of trigonometry and, in particular, the main trigonometric functions: sine, cosine and tangent

This Thematic Unit is part of our Mathematics collection

Cosine

Trigonometric ratio defined as the quotient between the side adjacent to an angle and the hypotenuse.

Degree

Angular measurement unit that divides a circle into 360 equal parts (°).

Radian

Angle measurement unit in the SI representing the central angle whose arc length is equal to the radius; it is approximately equal to 57.29°.

Sine

Trigonometric ratio defined as the quotient between the side opposite an angle and the hypotenuse.

Tangent

Trigonometric ratio defined as the quotient between the opposite side and the adjacent side to an angle.

Trigonometric Identity

A mathematical equality involving trigonometric functions that is true for any value of the angle.

Trigonometric Ratios

Proportions established between the sides of a right triangle with respect to one of its acute angles.

Trigonometry

Branch of mathematics that studies the relationships between the sides and angles of triangles.

Unit Circle

A circle with unit radius (r = 1) used to define the trigonometric functions of any angle.

What is trigonometry

Trigonometry is a branch of mathematics that focuses on the study of the relationships between the angles and sides of triangles. In particular, we study trigonometric functions, which are mathematical functions that relate the angles of a triangle to the lengths of its sides.

Basic concepts of trigonometry. Main trigonometric functions

The main trigonometric functions are sine, cosine and tangent. These functions are defined in terms of an acute angle within a right triangle.

Sine of an angle

The sine of an angle is equal to the length of the opposite side divided by the length of the hypotenuse.

Cosine of an angle

The cosine of an angle is equal to the length of the adjacent side divided by the length of the hypotenuse.

Tangent of an angle

The tangent of an angle is equal to the length of the opposite side divided by the length of the adjacent side.

Trigonometry Applications

Trigonometry has fundamental applications in various scientific, technical and practical areas:

Astronomy and navigation. It is used to calculate distances between planets, stars and determine geographic positions by measuring angles.
Engineering and architecture. It is essential in the design and construction of structures, bridges and buildings to ensure stability and accuracy in measurements.
Physics. It supports the study of wave phenomena, analysis of forces and motion, as well as optics and acoustics.
Cartography and geodesy. Helps to map and represent the earth’s surface and to measure distances and land areas.
Technology and computer graphics. It is used to model three-dimensional objects, generate graphics and visual effects in video games and animations.
Medicine and imaging. It is key in technologies such as tomography, magnetic resonance and ultrasound to interpret images of the human body.
Meteorology. Enables analysis of weather patterns and modeling of wind and storm trajectories.

Each of these applications demonstrates how trigonometry connects mathematics with the real world.

Cosine

Trigonometric ratio defined as the quotient between the side adjacent to an angle and the hypotenuse.

Degree

Angular measurement unit that divides a circle into 360 equal parts (°).

Radian

Angle measurement unit in the SI representing the central angle whose arc length is equal to the radius; it is approximately equal to 57.29°.

Sine

Trigonometric ratio defined as the quotient between the side opposite an angle and the hypotenuse.

Tangent

Trigonometric ratio defined as the quotient between the opposite side and the adjacent side to an angle.

Trigonometric Identity

A mathematical equality involving trigonometric functions that is true for any value of the angle.

Trigonometric Ratios

Proportions established between the sides of a right triangle with respect to one of its acute angles.

Trigonometry

Branch of mathematics that studies the relationships between the sides and angles of triangles.

Unit Circle

A circle with unit radius (r = 1) used to define the trigonometric functions of any angle.

Explore the exciting STEM world with our free, online, simulations and accompanying companion courses! With them you’ll be able to experience and learn hands-on. Take this opportunity to immerse yourself in virtual experiences while advancing your education – awaken your scientific curiosity and discover all that the STEM world has to offer!

Trigonometry simulations

Main trigonometric functions

This animation provides a representation of the variation of the sine, cosine and tangent as the angle changes.

When you are ready to start, click the “Resume” button.

Trigonometry on the Ferris wheel

This animation provides a representation of the sine function. Notice how the height of the red cab is determined by the sine function.

When you are ready to start click the “Go” button.

Trigonometry Units. Radians

This simulation allows to check how different angles are measured in radians.

Main trigonometric functions. Sine and cosine

With this simulation you can see how the sine and cosine functions are generated with the rotation of a segment.

Trigonometric Tour

This simulation will take you through a tour of the basic concepts of trigonometry using degrees or radians! Look for patterns in the values and in the graph when the value of theta is changed. Compare the graphs of sine, cosine, and tangent..

Giants of science

“If I have seen further, it is by standing on the shoulders of giants”

Isaac Newton

Archimedes

287 a.C.

–

212 a.C.

Archimedes established fundamental principles of mechanics and hydrostatics, including the famous Archimedes’ principle of buoyancy in fluids

“Give me a place to stand, and I will move the world”

Evangelista Torricelli

1608

–

1647

Evangelista Torricelli invented the mercury barometer, demonstrating the existence of atmospheric pressure, and made advances in hydrodynamics, geometry, and applied optics

“We have only today; let us make good use of this day”

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Giants of science

“If I have seen further, it is by standing on the shoulders of giants”

Isaac Newton

Johannes Kepler

1571

–

1630

Kepler formulated the three laws of planetary motion, establishing elliptical orbits and the relationship between planetary period and distance from the Sun

“The heavens teach the geometry that nature follows”

Euclides

300 a.C.

–

Euclid established the foundations of geometry, organizing axioms and theorems

“There is no royal road to geometry”

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Test your knowledge

What is trigonometry, and what does this branch of mathematics study?

Trigonometry is a branch of mathematics that studies the relationships between the angles and sides of triangles, especially right triangles. Its foundation lies in trigonometric functions, which connect an angle with the lengths of the sides that form it. The main functions are sine, cosine and tangent, defined from the proportions of the sides of a right triangle. Trigonometry not only describes these relationships but also provides tools to solve geometric problems, analyze periodic phenomena and model physical situations. Its importance extends to physics, engineering, astronomy and navigation, where distances, directions or force components must be calculated.

What are the main trigonometric functions, and how are they defined?

The fundamental trigonometric functions are sine, cosine and tangent, all defined from an acute angle in a right triangle. Sine is obtained by dividing the length of the opposite side by the hypotenuse, while cosine is the ratio of the adjacent side to the hypotenuse. Tangent is the quotient of the opposite side over the adjacent side. These functions relate angles to lengths and form the basis of many geometric and physical calculations. Their interpretation extends through the unit circle, where they are defined for any angle, enabling the study of periodic phenomena such as waves and vibrations.

What is trigonometry used for in everyday life?

Trigonometry helps solve problems involving distances, heights or directions without needing to reach the location physically. It is used in navigation, architecture, physics and technology. Computer graphics and video games rely on trigonometric functions to create 3D objects and realistic motion. Medical imaging techniques such as tomography and ultrasound also depend on trigonometric calculations.

How are sine, cosine and tangent related to the sides of a right triangle?

In a right triangle, each trigonometric function relates an angle to two of its sides. Sine is the opposite side divided by the hypotenuse, cosine is the adjacent side divided by the hypotenuse and tangent is the opposite side divided by the adjacent side.

In which fields is trigonometry applied, and why is it so useful?

Trigonometry is used in astronomy, navigation, engineering, architecture, physics, cartography and digital technology. It allows distances to be calculated, forces to be analyzed, structures to be designed and wave phenomena to be modeled. It is essential in computer graphics and medical imaging.