Trigonometry

Trigonometry () is a branch of mathematics concerned with relationships between angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. The Greeks focused on the calculation of chords, while mathematicians in India created the earliest-known tables of values for trigonometric ratios (also called trigonometric functions) such as sine.

triangle in which one angle is a 90-degree angle

thumb|right|A right-angled triangle and its hypotenuse

thumb|upright=1.4|right|A simplified illustration of the parallax of an object against a distant background due to a perspective shift. When viewed from "Viewpoint A", the object appears to be in front of the blue square. When the viewpoint is changed to "Viewpoint B", the object appears to have moved in front of the red square. thumb|right|This animation is an example of parallax. As the viewpoint moves side to side, the objects in the distance appear to move more slowly than the objects close to the camera. In this case, the white cube in front appears to move faster than the green cube in t

property of all triangles on a Euclidean plane

property of all triangles on a Euclidean plane

mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function

circle with radius one

function that repeats its values in regular intervals or periods

inverse function of the trigonometric function

mathematical curve that describes a smooth repetitive oscillation; continuous wave

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theorem

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thumb|right|200px |A circle of radius compressed to an ellipse. thumb|right|200px |A sphere of radius compressed to an oblate ellipsoid of revolution.

thumb|An example of series RLC circuit and respective phasor diagram for a specific . The arrows in the upper diagram are phasors, drawn in a phasor diagram ([[complex plane without axis shown), which must not be confused with the arrows in the lower diagram, which are the reference polarity for the voltages and the reference direction for the current.]]

mathematical curve outputted from a specific pair of parametric equations

measure for how wide an angle is

two equations relating the side lengths and angles of a triangle

theorem

function that relates the circular functions and hyperbolic functions without using complex numbers

angle of complex number about real axis

aspect of history

overview about the solution of triangles

quasi-periodic orbital trajectory

special function defined by an integral

sin² θ + cos² θ = 1

Simplification of the basic trigonometric functions

mathematical function

method for evaluating integrals involving trigonometric functions

CORDIC, short for coordinate rotation digital computer, is a simple and efficient algorithm to calculate trigonometric functions, hyperbolic functions, square roots, multiplications, divisions, exponentials, and logarithms with arbitrary base, typically converging with one digit (or bit) per iteration. CORDIC is therefore an example of a digit-by-digit algorithm. The original system is sometimes referred to as '''Volder's algorithm'''.

The only rational angles in first quadrant whose sine is rational are 0, 30 and 90 degrees

term in trigonometry

250px|right|thumb|The basic elements of Ptolemaic astronomy, showing a planet on an [[epicycle (smaller dashed circle), a deferent (larger dashed circle), the eccentric (×) and an equant (•).]]

theorem

overview about trigonometric tables

effect of filters or amplifiers on signals' phases as function of frequency

study of triangles in other spaces than the Euclidean plane

Prosthaphaeresis (from the Greek προσθαφαίρεσις) was an algorithm used in the late 16th century and early 17th century for approximate multiplication and division using formulas from trigonometry. For the 25 years preceding the invention of the logarithm in 1614, it was the only known generally applicable way of approximating products quickly. Its name comes from the Greek prosthen (πρόσθεν) meaning before and aphaeresis (ἀφαίρεσις), meaning taking away or subtraction.

irrational number produced by taking the sine or cosine of a rational multiple of a full circle

mathematical notation for cos(x) + i sin(x)

condition that must be fulfilled by a lens or other optical system in order for it to produce sharp images of off-axis as well as on-axis objects

reformulation of planar distance and angle measurements in terms of polynomial functions of Cartesian coordinates

famous mathematical optimization problem

overview about the trigonometric interpolation

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applications of trigonometry

parameter-free superresolution algorithm for the linear inverse problem in spectral estimation, direction-of-arrival estimation and tomographic reconstruction with applications in signal processing, medical imaging and remote sensing

2nd century CE trigonometric table

overview about mnemonics in trigonometry

magnification of angular error over distance