Trigonometry Definition

Trigonometry is a branch of geometry and mathematics that deals with the study of triangles, as well as the relationships that exist between their sides and their angles.

The triangle is a geometric figure that for hundreds of years has intrigued various civilizations, to the point of being considered by many to be the perfect figure. The development of trigonometry is the culmination of several centuries of study of these geometric figures. Although at first it seems that trigonometry is limited to the approach to triangles, the truth is that the concepts that it handles have applications in many areas of human knowledge. Trigonometry is used in distance measurement, signal analysis, mathematical modeling, etc.

The right triangle and the Pythagorean Theorem

To talk about trigonometry, perhaps we should start with the analysis of the right triangle. A right triangle is a type of triangle that is characterized by having a right angle, that is, an angle of 90 °.

In this right triangle, the sides \(a\) and \(b\) whose union forms the right angle are known as “legs”, the remaining side \(c\) and which is opposite the right angle is called “hypotenuse”. ”. There is a mathematical relationship between the three sides of the right triangle, which establishes that:
\({c^2} = {a^2} + {b^2}\)

This is the well-known “Pythagorean Theorem” which tells us that “in a right triangle the square of the hypotenuse is equal to the sum of the squares of the legs.”

The right triangle and the trigonometric functions

From the study of right triangles also arise the “Trigonometric Ratios” that relate the angles of a right triangle with its legs and its hypotenuse. The three main trigonometric ratios are the “sine”, the “cosine”, and the “tangent”. These trigonometric ratios for the angle \(\alpha \) of the triangle in the previous figure will be given by:

\(\sin \alpha = \frac{a}{c},\;\;\cos \alpha = \;\frac{b}{c},\;\;\tan \alpha = \frac{{\ sin \alpha }}{{\cos \alpha }} = \frac{a}{b}\)

For their part, these same trigonometric ratios for the angle \(\beta \) would be:

\(\sin \beta = \frac{b}{c},\;\;\cos \beta = \;\frac{a}{c},\;\;\tan \beta = \frac{{\ sin \beta }}{{\cos \beta }} = \frac{b}{a}\)

There are also other trigonometric ratios that depend on the above and are called “cosecant”, “secant” and “cotangent”. These trigonometric ratios for the angle \(\alpha \) would be given by:

\(\csc \alpha = \frac{1}{{\sin \alpha }} = \frac{c}{a},\;\;\sec \alpha = \frac{1}{{\cos \alpha }} = \frac{c}{b},\;\;\cot \alpha = \frac{1}{{\tan \alpha }} = \frac{b}{a}\)

All these trigonometric ratios relate the angles of a right triangle with quotients between its sides, however, we can extend the definition of these ratios to all real numbers and define the “trigonometric functions”.

The independent variable of a trigonometric function is an angle in radians, where 1 radian is the angle that spans an arc of circumference whose length is equal to the length of the radius and has a value of approximately 57.3°. The value of a trigonometric function at a given point is the value that the corresponding trigonometric ratio would have for that angle in radians.

Inverse Trigonometric Functions

At first the trigonometric ratios are very useful because thanks to these we can determine the measures of the angles of a right triangle if we know the measures of its sides, and vice versa.

There are functions that are inverses to the trigonometric functions mentioned above. The inverse functions can be defined as:

\(If\;y = \sin x ⟹ x = {\sin ^{ – 1}}y\)

\(If\;y = \cos x ⟹ x = {\cos ^{ – 1}}y\)

\(If\;y = \tan x ⟹ x = {\tan ^{ – 1}}y\)

The inverse trigonometric functions give us the value of the angle that corresponds to a certain value of its respective trigonometric ratio. For example, suppose that in the previous triangle we do not know the measure of the angle \(\alpha \), this angle can be obtained using the inverse trigonometric functions, in such a way that:

\(\alpha = {\sin ^{ – 1}}\frac{a}{c} = {\cos ^{ – 1}}\frac{b}{c} = {\tan ^{ – 1}} \frac{a}{b}\)

Applications of trigonometric functions

One of the most important applications of trigonometry is the measurement of distances. A clear example of this was the feat of the Greek astronomer and mathematician Eratosthenes who in the 3rd century BC realized that, during the solar noon of the summer solstice, objects in Siena did not cast a shadow, something contrary to what happened in Alexandria in that same moment where the objects did cast a shadow. Eratosthenes measured the size of the Earth using trigonometric functions, measuring the angle at which the sun’s rays fell on Alexandria at solar noon on the summer solstice and knowing the distance between Alexandria and Siena. Currently trigonometry is also used to measure the distance to nearby stars by means of the “stellar parallax” method.

The sine and cosine functions have a peculiarity, and that is that they have periodic behavior, that is, their values ​​are repeated with a certain frequency along the entire axis of the real numbers. This characteristic of the sine and cosine functions makes them perfect candidates for modeling phenomena that have oscillatory behavior such as waves, circular motion, planetary motion, alternating current circuits, etc.

In addition, trigonometry plays a fundamental role in the construction and analysis of signals. Any periodic signal can be decomposed as a sum of trigonometric functions, and vice versa. In this way, trigonometry helps us understand signals of all kinds, from those used in communications to those generated by electrophysiological processes in our body.

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