The branch of mathematics mainly concerned with relating the sides and angles of a triangle, based on triangles being similar if they have one right angle and one other angle equal. The trigonometric functions can be defined as the ratio of sides of a right-angled triangle, the commonest being 
. The most useful results for triangles that do not contain a right angle are the sine formula 
and the cosine formula a2 = b2 + c2 ? 2bc cos A.
Trigonometry (from the Greek trigonon = three angles and metron = measure ) is a branch of mathematics which deals with triangles, particularly triangles in a plane where one angle of the triangle is 90 degrees (right triangles). Trigonometry specifically deals with the relationships between the sides and the angles of triangles, that is, the trigonometric functions, and with calculations based on these functions.
Trigonometry has important applications in many branches of pure mathematics as well as of applied mathematics and, consequently, much of science.
Overview
Basic definitions
The shape of a right triangle is completely determined, up to similarity, by the value of either of the other two angles. These ratios are traditionally described by the following trigonometric functions of the known angle:
The sine function (sin), defined as the ratio of the leg opposite the angle to the hypotenuse. There are arithmetic relations between these functions, which are known as trigonometric identities.With these functions one can answer virtually all questions about arbitrary triangles by using the law of sines and the law of cosines.
The trigonometric functions can be defined in other ways besides the geometrical definitions above, using tools from calculus or infinite series.
Calculating trigonometric functions
Trigonometric functions were among the earliest uses for mathematical tables.
Today scientific calculators have buttons for calculating the main trigonometric functions (sin, cos, tan and sometimes cis) and their inverses. The floating point unit hardware incorporated into the microprocessor chips used in most personal computers have built in instructions for calculating trigonometric functions.
Early history of trigonometry
The origins of trigonometry can be traced to the civilizations of ancient Egypt, Mesopotamia and the Indus Valley, more than 4000 years ago.
Some experts believe that trigonometry was originally invented to calculate sundials, a traditional exercise in the oldest books.
The first recorded use of trigonometry came from the Hellenistic mathematician Hipparchus circa 150 BC, who compiled a trigonometric table using the sine for solving triangles.
The ancient Sinhalese, when constructing reservoirs in the Anuradhapura kingdom, used trigonometry to calculate the gradient of the water flow.
The Indian mathematician Aryabhata in 499, gave tables of half chords which are now known as sine tables, along with cosine tables.
Another Indian mathematician, Brahmagupta in 628, used an interpolation formula to compute values of sines, up to the second order of the Newton-Stirling interpolation formula.
In the 10th century, the Persian mathematician and astronomer Abul Wáfa introduced the tangent function and improved methods of calculating trigonometry tables. sin (a + b), and discovered the sine formula for spherical geometry:
Also in the late 10th and early 11th centuries, the Egyptian astronomer Ibn Yunus performed many careful trigonometric calculations and demonstrated the formula cos(a)cos(b) = (1 / 2)[cos(a + b) + cos(a − b)].
Indian mathematicians were the pioneers of variable computations algebra for use in astronomical calculations along with trigonometry.
Persian mathematician Omar Khayyám (1048-1131) combined trigonometry and approximation theory to provide methods of solving algebraic equations by geometrical means.
Detailed methods for constructing a table of sines for any angle were given by the Indian mathematician Bhaskara in 1150, along with some sine and cosine formulae.
The 13th century Persian mathematician Nasir al-Din Tusi, along with Bhaskara, was probably the first to treat trigonometry as a distinct mathematical discipline. Nasir al-Din Tusi in his Treatise on the Quadrilateral was the first to list the six distinct cases of a right angled triangle in spherical trigonometry.
In the 14th century, Persian mathematician al-Kashi and Timurid mathematician Ulugh Beg (grandson of Timur) produced tables of trigonometric functions as part of their studies of astronomy.
The mathematician Bartholemaeus Pitiscus published an influential work on trigonometry in 1595 which may have coined the word "trigonometry".
Applications of trigonometry
There are an enormous number of applications of trigonometry and trigonometric functions.
Fields which make use of trigonometry or trigonometric functions include astronomy (especially, for locating the apparent positions of celestial objects, in which spherical trigonometry is essential) and hence navigation (on the oceans, in aircraft, and in space), music theory, acoustics, optics, analysis of financial markets, electronics, probability theory, statistics, biology, medical imaging (CAT scans and ultrasound), pharmacy, chemistry, number theory (and hence cryptology), seismology, meteorology, oceanography, many physical sciences, land surveying and geodesy, architecture, phonetics, economics, electrical engineering, mechanical engineering, civil engineering, computer graphics, cartography, crystallography and game development.
Common formulae
Certain equations involving trigonometric functions are true for all angles and are known as trigonometric identities. Many express important geometric relationships.
Trigonometric identities
Pythagorean identities
Sum and difference identities
Triangle identities
Law of sines
The law of sines for an arbitrary triangle states:
or equivalently:
Law of cosines
The law of cosines (also known as the cosine formula) is an extension of the Pythagorean theorem to arbitrary triangles:
c + b2 − 2abcosC,or equivalently:
Law of tangents
The law of tangents:
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