Translation and analysis of words by ChatGPT artificial intelligence
On this page you can get a detailed analysis of a word or phrase, produced by the best artificial intelligence technology to date:
- how the word is used
- frequency of use
- it is used more often in oral or written speech
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- usage examples (several phrases with translation)
- etymology
What (who) is logarithmic cosine - definition
![proportional]].](https://commons.wikimedia.org/wiki/Special:FilePath/Academ Base of trigonometry.svg?width=200 "proportional]].")
![A [[Lissajous curve]], a figure formed with a trigonometry-based function.](https://commons.wikimedia.org/wiki/Special:FilePath/Lissajous curve 5by4.svg?width=200 "A [[Lissajous curve]], a figure formed with a trigonometry-based function.")
![k}} is large is called the [[Gibbs phenomenon]]](https://commons.wikimedia.org/wiki/Special:FilePath/Sawtooth Fourier Animation.gif?width=200 "k}} is large is called the [[Gibbs phenomenon]]")
![An animation of the [[additive synthesis]] of a [[square wave]] with an increasing number of harmonics](https://commons.wikimedia.org/wiki/Special:FilePath/Synthesis square.gif?width=200 "An animation of the [[additive synthesis]] of a [[square wave]] with an increasing number of harmonics")
![Taylor polynomial]] of degree 7 (pink) for a full cycle centered on the origin.](https://commons.wikimedia.org/wiki/Special:FilePath/Taylorsine.svg?width=200 "Taylor polynomial]] of degree 7 (pink) for a full cycle centered on the origin.")
![Cotangent (dotted)}}
– [{{filepath:trigonometric_functions_derivation_animation.svg}} animation]](https://commons.wikimedia.org/wiki/Special:FilePath/Trigonometric functions.svg?width=200 "Cotangent (dotted)}}
– [{{filepath:trigonometric_functions_derivation_animation.svg}} animation]")
![The [[unit circle]], with some points labeled with their cosine and sine (in this order), and the corresponding angles in radians and degrees.](https://commons.wikimedia.org/wiki/Special:FilePath/Unit circle angles color.svg?width=200 "The [[unit circle]], with some points labeled with their cosine and sine (in this order), and the corresponding angles in radians and degrees.")
 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg?width=200)
Wikipedia
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis.
The trigonometric functions most widely used in modern mathematics are the sine, the cosine, and the tangent. Their reciprocals are respectively the cosecant, the secant, and the cotangent, which are less used. Each of these six trigonometric functions has a corresponding inverse function, and an analog among the hyperbolic functions.
The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles. To extend the sine and cosine functions to functions whose domain is the whole real line, geometrical definitions using the standard unit circle (i.e., a circle with radius 1 unit) are often used; then the domain of the other functions is the real line with some isolated points removed. Modern definitions express trigonometric functions as infinite series or as solutions of differential equations. This allows extending the domain of sine and cosine functions to the whole complex plane, and the domain of the other trigonometric functions to the complex plane with some isolated points removed.
