what is named this sine mathematics U _U
Answers
Answer:
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle (the hypotenuse), and the cosine is the ratio of the length of the adjacent leg to that of the hypotenuse. For an angle {\displaystyle x} x, the sine and cosine functions are denoted simply as {\displaystyle \sin x} \sin x and {\displaystyle \cos x} \cos x.[1]
Sine and cosine
Sine cosine one period.svg
General information
General definition
{\displaystyle {\begin{aligned}&\sin(\alpha )={\frac {\textrm {opposite}}{\textrm {hypotenuse}}}\\[8pt]&\cos(\alpha )={\frac {\textrm {adjacent}}{\textrm {hypotenuse}}}\\[8pt]\end{aligned}}} {\displaystyle {\begin{aligned}&\sin(\alpha )={\frac {\textrm {opposite}}{\textrm {hypotenuse}}}\\[8pt]&\cos(\alpha )={\frac {\textrm {adjacent}}{\textrm {hypotenuse}}}\\[8pt]\end{aligned}}}
Motivation of invention
Indian astronomy
Date of solution
Gupta period
Fields of application
Trigonometry, integral transform, etc.
Domain and Range
Domain
(−∞, +∞) a
Codomain
[−1, 1] a
Basic features
Parity
Sine: odd; cosine: even
Period
2π
Specific values
At zero
Sine: 0; cosine: 1
Maxima
Sine: (2kπ +
π
/
2
, 1)b; cosine: (2kπ, 1)
Minima
Sine: (2kπ −
π
/
2
, −1); cosine: (2kπ + π, -1)
Specific features
Root
Sine: kπ; cosine: kπ +
π
/
2
Critical point
Sine: kπ +
π
/
2
; cosine: kπ
Inflection point
Sine: kπ; cosine: kπ +
π
/
2
Fixed point
Sine: 0; cosine: Dottie number
Related functions
Reciprocal
Sine: cosecant; cosine: secant
Inverse
Sine: arcsine; cosine: arccosine
Derivative
{\displaystyle {\begin{aligned}{\frac {d}{dx}}\sin(x)&=\cos(x)\\[8pt]{\frac {d}{dx}}\cos(x)&=-\sin(x)\\[8pt]\end{aligned}}} {\displaystyle {\begin{aligned}{\frac {d}{dx}}\sin(x)&=\cos(x)\\[8pt]{\frac {d}{dx}}\cos(x)&=-\sin(x)\\[8pt]\end{aligned}}}
Antiderivative
{\displaystyle {\begin{aligned}\int \sin(x)\,dx&=-\cos(x)+C\\[8pt]\int \cos(x)\,dx&=\sin(x)+C\\[8pt]\end{aligned}}} {\displaystyle {\begin{aligned}\int \sin(x)\,dx&=-\cos(x)+C\\[8pt]\int \cos(x)\,dx&=\sin(x)+C\\[8pt]\end{aligned}}}
Other Related
tan, csc, sec, cot
Series definition
Taylor series
{\displaystyle {\begin{aligned}\sin(x)&=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots \\[8pt]&={}\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}x^{2n+1}\\[8pt]\cos(x)&=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots \\[8pt]&={}\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}x^{2n}\\[8pt]\end{aligned}}} {\displaystyle {\begin{aligned}\sin(x)&=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots \\[8pt]&={}\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}x^{2n+1}\\[8pt]\cos(x)&=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots \\[8pt]&={}\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}x^{2n}\\[8pt]\end{aligned}}}
a For real numbers.
b Variable k is an integer.
More generally, the definitions of sine and cosine can be extended to any real value in terms of the lengths of certain line segments in a unit circle. More modern definitions express the sine and cosine as infinite series, or as the solutions of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers.
The sine and cosine functions are commonly used to model periodic phenomena such as sound and light waves, the position and velocity of harmonic oscillators, sunlight intensity and day length, and average temperature variations throughout the year.
The functions sine and cosine can be traced to the jyā and koṭi-jyā functions used in Gupta period Indian astronomy (Aryabhatiya, Surya Siddhanta), via translation from Sanskrit to Arabic, and then from Arabic to Latin.[2] The word "sine" (Latin "sinus") comes from a Latin mistranslation by Robert of Chester of the Arabic jiba, which is a transliteration of the Sanskrit word for half the chord, jya-ardha.[3] The word "cosine" derives from a contraction of the Medieval Latin "complementi sinus".