Original filename: formule Title: Author: Raffaele SANTORO. This PDF document has been generated by Microsoft®. p2hv{}erlappingourstyleplease{position:absolute;top:0;left:0;bottom :0;widthpx;padding:0;margin:0;overflow:auto}.p2hv. Trigonometry is a branch of mathematics that studies relationships involving lengths and . Bartholomaeus Pitiscus was the first to use the word, publishing his Trigonometria in .. The law of cosines (known as the cosine formula, or the “cos rule”) is an extension of the Pythagorean theorem to arbitrary triangles .

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The field emerged in the Hellenistic world during the 3rd century Trigonnometriche from applications of geometry to astronomical studies. The 3rd-century astronomers first noted that the lengths of the sides of a right-angle triangle trigonometricye the tritonometriche between those sides have fixed relationships: These calculations soon came to be defined as the trigonometric functions and trigonoetriche are pervasive in both pure and applied mathematics: Trigonometry is also the foundation of surveying.

Trigonometry is most trionometriche associated with planar right-angle triangles each of which is a two-dimensional triangle with one angle equal to 90 degrees. The applicability to non-right-angle triangles exists, but, since any non-right-angle triangle on a flat plane can be bisected to create two right-angle triangles, most problems can be reduced to calculations trigononetriche right-angle triangles.

Thus the majority of applications relate to right-angle triangles. One exception to this trigoonmetriche spherical trigonometrythe study of triangles on spheressurfaces of constant positive curvaturein elliptic geometry a fundamental part of astronomy and navigation.

Trigonometry on surfaces of negative curvature is part of hyperbolic geometry. Trigonometry basics are often taught in schools, either as a separate course or as a part of a precalculus course. Sumerian astronomers studied angle measure, using a division of circles into degrees.

The ancient Nubians used a similar method. In the 3rd century BC, Hellenistic mathematicians such as Euclid and Archimedes studied the properties of chords and inscribed angles in circles, and they proved theorems that are trigonommetriche to modern trigonometric formulae, although they presented them geometrically rather than algebraically. In BC, Hipparchus from NicaeaAsia Minor gave the first tables of chords, analogous to modern tables of sine valuesand used fromule to solve problems in trigonometry and spherical trigonometry.

Centuries passed before more detailed tables were produced, and Ptolemy’s treatise remained in use for performing trigonometric calculations in astronomy throughout the next years in the medieval ByzantineIslamictribonometriche, later, Western European worlds. The modern sine convention is first attested in the Surya Siddhantaand its properties were further documented by the 5th century AD Indian mathematician and astronomer Aryabhata.

By the 10th century, Islamic mathematicians were using all six trigonometric functions, had tabulated their values, and were applying them to problems in spherical geometry. The Persian polymath Nasir al-Din al-Tusi has been described as the creator of trigonometry as a mathematical discipline in its own right. Driven by the demands of navigation and the growing need for accurate maps of large geographic areas, trigonometry grew into a major branch of mathematics.

It was Leonhard Euler who fully incorporated complex numbers into trigonometry. The works of the Scottish mathematicians James Gregory in the 17th century and Colin Maclaurin in the 18th century were influential in the development trigonometrifhe trigonometric series.

If one angle of a triangle is 90 degrees and one of the other angles is known, the third is thereby fixed, because the three angles of any triangle add up to degrees. The two acute angles therefore add up to 90 degrees: The shape of a triangle is completely determined, except for similarityby the angles.

Once the angles are known, formupe ratios of the sides are determined, regardless of the overall size of the triangle. If the length of one of the sides is known, the other two are determined. These ratios are given by the following trigonometric functions of the known angle Awhere ab and c refer to the lengths of the sides in the accompanying figure:. The hypotenuse is the side opposite to the 90 degree angle in a right triangle; it is the longest side of the triangle and one of the two sides adjacent to angle A.

The adjacent leg is the other side that is adjacent to angle A. The opposite side is the trigonometrlche that is opposite to angle A.

The terms perpendicular and base are sometimes used for the opposite and adjacent sides respectively. The formuel of these functions are named the cosecant cscsecant secand cotangent cotrespectively:.

### Trigonometria – Goniometria

The inverse functions are called the arcsinearccosineand arctangentrespectively. There are arithmetic relations between these functions, which are known as trigonometric identities. The cosine, cotangent, and cosecant are so named because they are respectively the sine, tangent, and secant of the complementary angle abbreviated to “co-“.

With these functions, one can answer virtually fornule questions about arbitrary triangles by using the law of sines and the law of cosines.

These laws can be used to compute the remaining angles and sides of any triangle as soon as two sides and their included angle or two angles and a side or three sides are known. These laws are useful in all branches of geometry, since every polygon may be described as a finite combination of triangles.

Using the unit circleone can extend them to all positive and negative arguments see trigonometric function. That means their values repeat at those intervals. The trigonometric functions can be defined in other ways besides the geometrical definitions above, using tools from calculus and infinite series.

With these definitions the trigonometric functions can be defined for complex numbers. The complex exponential function is particularly useful. See Euler’s and De Moivre’s formulas. A common use of mnemonics is to remember facts and relationships in trigonometry. For example, the sinecosineand tangent ratios in a right triangle can be remembered by representing them and their corresponding sides as strings of letters.

One way to remember the letters is to sound them out phonetically i. Trigonometric functions were among the earliest uses for mathematical tables. Such tables were incorporated into mathematics textbooks and students were taught to look up values and how to interpolate between the values listed to get higher accuracy.

### Penna con le Formule

Slide rules had special scales for trigonometric functions. Today, scientific calculators have buttons for calculating the main trigonometric functions sin, cos, tan, and sometimes cis and their inverses.

Most allow a choice of angle measurement methods: Most computer programming languages provide function libraries that include the trigonometric functions.

The floating point unit hardware incorporated into the microprocessor chips used in most personal computers has built-in instructions for calculating trigonometric functions. There is an enormous number of uses of trigonometry and trigonometric functions. For instance, the technique of triangulation is used in astronomy to measure the distance to nearby stars, in geography to measure distances between landmarks, and in satellite navigation systems. The sine and cosine functions are fundamental to the theory of periodic functionssuch as those that describe sound and light waves.

Fields that use trigonometry or trigonometric functions include astronomy especially for locating apparent positions of celestial objects, in which spherical trigonometry is essential and hence navigation on the oceans, in aircraft, and in spacemusic theoryaudio synthesisacousticsopticselectronicsbiologymedical imaging CT scans and ultrasoundpharmacychemistrynumber theory and hence cryptologyseismologymeteorologyoceanographymany physical sciencesland surveying and geodesyarchitectureimage compressionphoneticseconomicselectrical engineeringmechanical engineeringcivil engineeringcomputer rrigonometrichecartographycrystallography and game development.

The following identities are related to the Pythagorean theorem and hold for any value: Certain equations involving trigonometric functions are true for all angles and are known as trigonometric identities. Some identities equate an expression to a different expression involving the rormule angles.

These are listed in List of trigonometric identities. Triangle identities that relate the sides and angles of a given triangle are listed below.

In the following identities, A formulw, B and C are the angles of a triangle and ab and c are the lengths of sides of the triangle opposite the respective angles as shown in the diagram.

The law of sines also known as the “sine rule” for an arbitrary triangle states:. Another law involving sines can be used trjgonometriche calculate the area of a triangle.

Given two sides a and b and the angle between the sides Cthe area of the triangle is given by half the product of the lengths of two sides and the sine of the angle between the two sides:. The law of cosines known as the cosine formula, or the “cos rule” is an extension of the Pythagorean theorem to arbitrary triangles:.

The law of cosines may be used to prove Heron’s formulawhich is another method that may be used to calculate the area of a triangle.

## Trigonometry

This formula states that if a triangle has sides of lengths aband cand if the semiperimeter is. From Wikipedia, the free encyclopedia. For other uses, see Trig disambiguation.

This section does not cite any sources. Please help improve this section by adding citations to reliable sources. Unsourced material may be challenged and removed. March Learn how and when to remove this template message. A History of Mathematics. Episodes trigonometrche the Early History of Astronomy.

A history of ancient mathematical astronomy. One of al-Tusi’s most important mathematical contributions was the creation of trigonometry as a mathematical discipline in its own right trigonometrichd than as just a tool for astronomical applications. In Treatise on the quadrilateral al-Tusi gave the first extant exposition of the whole system of plane fotmule spherical trigonometry. This work is really the first in history on trigonometry as an independent branch of pure mathematics and the first in which all six cases for a right-angled spherical triangle are set forth.

Check date values in: Biography — Encyclopaedia Iranica”. His major contribution in mathematics Nasr,pp. Spherical trigonometry also owes its development to his efforts, and this includes the concept of the six fundamental formulas for the solution of spherical right-angled triangles. From Byzantium to Italy.

Greek Studies in the Italian RenaissanceLondon.

The Rainbow of Mathematics: A History of the Mathematical Sciences. From Kant to Hilbert: Oxford University Press US. Focus on Curves and Surfaces. A Very Short Introduction. Technical Mathematics with Calculus illustrated ed. Extract of page Trigonometry at Wikipedia’s sister projects.

Linear Multilinear Abstract Elementary. Philosophy of mathematics Mathematical logic Set theory Category trigonometricbe.