Trigonometry - Basic concepts

The trigonometry (whose term derives from the Greek ¤ä¤ü╬╣╬│¤ë╬Ż╬┐¤Ť trig┼Źnos "triangle" and ╬╝╬Á¤ä¤ü╬┐╬Ż metron "measure") is a branch of mathematics that deals with the study of angles, triangles and trigonometric ratios: sine, cosine, tangent, cotangent, secant and cosecant. Although it was in the 17th century that trigonometry was incorporated into mathematical analysis, its real origins go back to the ancient Egyptian pyramids and Babylonian astronomy.

Brief history of trigonometry

The first studies of triangles date back to 2000 BC; The ancient Egyptians and Babylonians knew the theorems about the proportions of the sides of similar triangles, however, since they lacked the concept of an angular measure, they limited themselves to studying the sides of the triangles. Subsequently, the ancient Greek and Hellenistic mathematicians began to make use of the chord, which is defined geometrically as in Figure 1. The chord of an angle is the dimensional length of a chord between two points in a circular unit separated by an angle.

Rope
Figure 1. The chord of an angle is a semirect whose ends are over the arch.

In the 3rd century BC, Greek mathematicians and philosophers such as Euclid and Archimedes studied the properties of chords and angles inscribed in circles, and proved theorems that are equivalent to modern trigonometric formulas, although they presented them geometrically rather than algebraically.

In 140 BC, the Greek astronomer and mathematician Hipparchus of Nicea (190 BC - 120 BC) constructed a "string table" analogous to modern breast tables and used them to solve problems in trigonometry and spherical trigonometry. For this reason he is considered the founder of trigonometry. In the 2nd century AD. C., the Greek-Egyptian astronomer Ptolemy built detailed trigonometric tables, which were used for the next 1200 years to perform trigonometric calculations in astronomy.

The modern sine convention is attested for the first time in the Suria-siddhanta (one of the first books on archaeoastronomy of the Hinduists), and its properties were documented by the Indian mathematician and astronomer Aryabhata (5th century AD). Both Indian and Greek works were translated and expanded by medieval Islamic mathematicians such as Al-Juarismi and Abú al-Wafá. Due to these studies, trigonometry became an independent discipline in the Islamic world. By the 10th century, Islamic mathematicians knew the six trigonometric functions, tabulated their values, and applied them to problems.

Beginning in the Renaissance, translations of Arabic and Greek texts led the West to become familiar with Arabic trigonometry. The first major paper on this subject was written by the German mathematician and astronomer Johannes M├╝ller von K├Ânigsberg, known as Regiomontano.

Modern trigonometry

The study of modern trigonometry begins during the Enlightenment, beginning with Isaac Newton and James Stirling, and the development of the general Newton-Stirling interpolation formula for trigonometric functions; and it reaches its modern form with the Swiss mathematician Leonhard Euler. Euler with his Introductio in analysis without infinitorum (1748) was responsible for establishing the analytical treatment of trigonometric functions in Europe, deriving its infinite series and presenting the "Euler's Formula". He also used abbreviations very similar to those we currently know (sin; cos; tang; cot; sec; y cosec).

Before this, the English mathematician Roger Cotes had calculated the derivative of the sine in his Harmony Mensurarum (1722). Also in the 18th century, the also English mathematician Brook Taylor defined the theorem that bears his name and gave the series expansions and approximations for the six trigonometric functions. The works of Scottish mathematicians James Gregory in the 17th century and Colin Maclaurin in the 18th century were also highly influential in the development of trigonometric series.

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Definition of trigonometry

Trigonometry is the study of the relationship between the sides and angles of a right triangle and a circumference. Due to this, a series of functions were defined, called trigonometric ratios: sine, cosine, tangent, cotangent, secant and cosecant.

Trigonometry application

Trigonometry is applied in other branches of mathematics, such as geometry, also in physics, as such is the case with mechanics. It has numerous applications in astronomy, cartography, nautical, telecommunications, architecture, topography, engineering, medicine, and much more. For example, in astronomy it is used to measure distances to nearby satellites, planets, and stars; in architecture for the creation of buildings, in engineering for the construction of bridges and in topography for surveying land. It is used in medicine in the graphic recording of the electrical activity of the heart (electrocardiograms), as well as in others.

Applications of trigonometry

Angular units

In the measurement of angles and, consequently, in trigonometry, three units are used: radian, sexagesimal degree and centesimal degree. Although in our daily life we make use of the sexagesimal degree, in mathematics the radian is used; in architecture, topography and construction the centesimal degree is used.

Radian

It is the unit of measurement of an angle with a vertex in the center of a circle whose sides are cut by the arc of the circle; This arc has a length equal to that of the radius. The unit is written 1 rad. One full turn of the circumference is 2¤Ç rad. Radians are the only angle measurement unit in the International System of Units.

Sexagesimal degree

Angle unit that divides a circle into 360 degrees. The unit is written 1st.

Each degree is divided into 60 minutes (') and each minute into 60 seconds (┬╗), or what is the same, 1 ┬░ = 60 ÔÇ▓ and 1' = 60 ÔÇ│.

Centesimal degree

Angular unit that divides a circumference into 400 centesimal degrees. The unit is written 1g. It is also called gon, gradián and gonio. Each centesimal degree is divided into 100 centesimal minutes, and this, in turn, into 100 centesimal seconds.

The relationships between the three units are:

Relations between angular units
Figure 2. Relations between angular units

Trigonometric ratios

The trigonometric ratios of a right angle are: sine, cosine, tangent, cosecant, secant, and cotangent.

Trigonometric ratios

Sine of an angle a

The sine is the quotient of the leg opposite the angle between the hypotenuse:

\ LARGE \ sin A = \ frac {a} {c}

Cosine of an angle a

The cosine is the quotient of the leg contiguous to the angle between the hypotenuse:

\ LARGE \ cos A = \ frac {b} {c}

Tangent of an angle a

The tangent is the quotient of the opposite leg between the adjacent leg.

\ LARGE \ tan A = \ frac {a} {b}

In the same way, we deduce that it is also the quotient of the sine between the cosine of an angle:

\ LARGE \ tan A = \ frac {\ sin A} {\ cos A}

Tangent can also be abbreviated as tg.

╬Ĺ Cosecant

The cosecant of A is the reciprocal trigonometric ratio to the sine (with the inverted quotient):

\ LARGE \ csc A = \ frac {1} {\ sin A}

╬Ĺ blotter

The secant of A is the reciprocal trigonometric ratio to the cosine (with the inverted quotient):

\ LARGE \ sec A = \ frac {1} {\ cos A}

Cotangent of ╬▒

The cotangent of A is the reciprocal trigonometric ratio to the tangent (with the inverted quotient):

\ LARGE \ cot A = \ frac {1} {\ tan A}

Graphic representations

Graphical representations sine, cosine and tangent
Figure 3.Sine, cosine, and tangent graphical representations
Graphic representations secant, cosecant and cotangent
Figure 4. Graphic representations secant, cosecant and cotangent.

Trigonometric identities

A trigonometric identity is an equality that links two trigonometric functions and is valid for the values of the angles (variables) in which it is defined. From Pythagoras theorem we can derive the fundamental or basic identities and from these, others called auxiliaries.

The main formulas e trigonometric identities are:

Basic identities

\ large \ begin {align *} & \ sin \ theta \ cdot \ csc \ theta = 1 \\ & \ tan \ theta \ cdot \ cot \ theta = 1 \\ & \ cos \ theta \ cdot \ sec \ theta = 1 \\ \ end {align *}

Pythagorean identities

\ large \ begin {align *} & \ sin ^ {2} \ theta \ cdot \ cos ^ {2} \ theta = 1 \\ & 1 + \ tan ^ {2} \ theta = \ sec ^ {2} \ theta \ \ & 1 + \ cot ^ {2} \ theta = \ csc ^ {2} \ theta \\ \ end {align *}

Odd and even identities

Even functions:

\ large \ begin {align *} & \ cos \ left (- \ theta \ right) = \ cos \ theta \\ & \ sec \ left (- \ theta \ right) = \ sec \ theta \ end {align *}

Odd functions

\ large \ begin {align *} & \ sin \ left (- \ theta \ right) = - \ sin \ theta \\ & \ csc \ left (- \ theta \ right) = - \ csc \ theta \\ & \ tan \ left (- \ theta \ right) = - \ tan \ theta \\ & \ cot \ left (- \ theta \ right) = - \ cot \ theta \ end {align *}

Formulas for trigonometric functions of addition and subtraction of angles

\ large \ begin {align *} & \ sin \ left (\ alpha \ pm \ beta \ right) = \ sin \ alpha \ cdot \ cos \ beta \ pm \ sin \ beta \ cdot \ cos \ alpha \\ & \ cos \ left (\ alpha \ pm \ beta \ right) = \ cos \ alpha \ cdot \ cos \ beta \ pm \ sin \ beta \ cdot \ sin \ alpha \\ & \ tan \ left (\ alpha \ pm \ beta \ right) = \ frac {\ tan \ alpha \ pm \ tan \ beta} {1 \ pm \ tan \ alpha \ cdot \ tan \ beta} \ end {align *}

Formulas for double angles

\ large \ begin {align *} & \ sin 2 \ theta = 2 \ cdot \ sin \ theta \ cdot \ cos \ theta \\ & \ bullet \ cos2 \ theta = 2 \ cdot \ cos ^ {2} \ theta - \ sin ^ {2} \ theta \\ & \ bullet \ cos 2 \ theta = 1 - 2 \ cdot \ sin ^ {2} \ theta \\ & \ bullet \ cos 2 \ theta = 2 \ cdot \ cos ^ { 2} \ theta -1 \\ & \ tan 2 \ theta = \ frac {2 \ cdot \ tan \ theta} {1- \ tan ^ {2} \ theta} \ end {align *}

Formulas for average angles

\ large \ begin {align *} & \ sin \ left (\ frac {\ theta} {2} \ right) = \ pm \ sqrt {\ frac {1- \ cos \ theta} {2}} \\ & \ cos \ left (\ frac {\ theta} {2} \ right) = \ pm \ sqrt {\ frac {1+ \ cos \ theta} {2}} \\ & \ tan \ left (\ frac {\ theta} {2} \ right) = \ pm \ sqrt {\ frac {1- \ cos \ theta} {1+ \ cos \ theta}} = \ frac {1- \ cos \ theta} {\ sin \ theta} = \ frac {\ sin \ theta} {1+ \ cos \ theta} \\ & \ end {align *}

Product identities - sum

\ large \ begin {align *} & \ sin \ alpha \ cdot \ sin \ beta = \ frac {1} {2} \ left [\ cos \ left (\ alpha - \ beta \ right) - \ cos \ left ( \ alpha + \ beta \ right) \ right] \\ & \ cos \ alpha \ cdot \ cos \ beta = \ frac {1} {2} \ left [\ cos \ left (\ alpha - \ beta \ right) + \ cos \ left (\ alpha + \ beta \ right) \ right] \\ & \ sin \ alpha \ cdot \ cos \ beta = \ frac {1} {2} \ left [\ sin \ left (\ alpha + \ beta \ right) + \ sin \ left (\ alpha - \ beta \ right) \ right] \ end {align *}

Sum - Product Identities

\ large \ begin {align *} & \ sin \ alpha + \ sin \ beta = 2 \ cdot \ sin \ left (\ frac {\ alpha + \ beta} {2} \ right) \ cdot \ cos \ left (\ frac {\ alpha - \ beta} {2} \ right) \\ & \ sin \ alpha - \ sin \ beta = 2 \ cdot \ sin \ left (\ frac {\ alpha - \ beta} {2} \ right) \ cdot \ cos \ left (\ frac {\ alpha + \ beta} {2} \ right) \\ & \ cos \ alpha + \ cos \ beta = 2 \ cdot \ cos \ left (\ frac {\ alpha + \ beta} {2} \ right) \ cdot \ cos \ left (\ frac {\ alpha - \ beta} {2} \ right) \\ & \ cos \ alpha - \ cos \ beta = - 2 \ cdot \ sin \ left (\ frac {\ alpha + \ beta} {2} \ right) \ cdot \ sin \ left (\ frac {\ alpha - \ beta} {2} \ right) \ end {align *}