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{{other uses}}{{distinguish|sign|sign (mathematics)}}

}}, +{{lower{{resize∞}}}}) {{smallsupcodomain=[−1, 1] {{smallsupperiod=2{{pi}}plusinf= |minusinf= k{{pi}} + {{sfrac>{{pi}}b}} k{{pi}} − {{sfrac>{{pi}}|2}}, −1)f1= f5=root=k{{pi}} k{{pi}} + {{sfrac>{{pi}}inflection=k{{pi}} |fixed=0{{supReal number>real numbers. b}} Variable k is an integer.}}}}{{Trigonometry}}In mathematics, the sine is a trigonometric function of an angle. The sine of an acute angle is defined in the context of a right triangle: for the specified angle, it 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).More generally, the definition of sine (and other trigonometric functions) can be extended to any real value in terms of the length of a certain line segment in a unit circle. More modern definitions express the sine as an infinite series or as the solution of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers.The sine function is 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 function sine can be traced to the jyā and koá¹­i-jyā functions used in Gupta period Indian astronomy (Aryabhatiya, Surya Siddhanta), via translation from Sanskrit to Arabic and then from Arabic to Latin.Uta C. Merzbach, Carl B. Boyer (2011), A History of Mathematics, Hoboken, N.J.: John Wiley & Sons, 3rd ed., p. 189. The word "sine" (Latin "sinus") comes from a Latin mistranslation of the Arabic jiba, which is a transliteration of the Sanskrit word for half the chord, jya-ardha.Victor J. Katz (2008), A History of Mathematics, Boston: Addison-Wesley, 3rd. ed., p. 253, sidebar 8.1. WEB,weblink Archived copy, 2015-04-09, no,weblink" title="">weblink 2015-04-14,

Right-angled triangle definition

(File:Trigono sine en2.svg|right|thumb| For the angle α, the sine function gives the ratio of the length of the opposite side to the length of the hypotenuse.)To define the sine function of an acute angle α, start with a right triangle that contains an angle of measure α; in the accompanying figure, angle A in triangle ABC is the angle of interest. The three sides of the triangle are named as follows:
  • The opposite side is the side opposite to the angle of interest, in this case side a.
  • The hypotenuse is the side opposite the right angle, in this case side h. The hypotenuse is always the longest side of a right-angled triangle.
  • The adjacent side is the remaining side, in this case side b. It forms a side of (is adjacent to) both the angle of interest (angle A) and the right angle.
Once such a triangle is chosen, the sine of the angle is equal to the length of the opposite side divided by the length of the hypotenuse, or:
sin(alpha) = frac {textrm{opposite}} {textrm{hypotenuse}}
The other trigonometric functions of the angle can be defined similarly; for example, the cosine of the angle is the ratio between the adjacent side and the hypotenuse, while the tangent gives the ratio between the opposite and adjacent sides.As stated, the value sin(alpha) appears to depend on the choice of right triangle containing an angle of measure α. However, this is not the case: all such triangles are similar, and so the ratio is the same for each of them.

Unit circle definition

File:Unit circle.svg|right|thumb|Unit circle: the radius has length 1. The variable t measures the angleangleIn trigonometry, a unit circle is the circle of radius one centered at the origin (0, 0) in the Cartesian coordinate system.Let a line through the origin, making an angle of θ with the positive half of the x-axis, intersect the unit circle. The x- and y-coordinates of this point of intersection are equal to {{math|cos(θ)}} and {{math|sin(θ)}}, respectively. The point's distance from the origin is always 1.Unlike the definitions with the right triangle or slope, the angle can be extended to the full set of real arguments by using the unit circle. This can also be achieved by requiring certain symmetries and that sine be a periodic function.File:Circle_cos_sin.gif|thumb|450px|left|Animation showing how the sine function (in red) y = sin(theta) is graphed from the y-coordinate (red dot) of a point on the unit circleunit circle{{clr}}


Exact identities (using radians):These apply for all values of theta.
sin(theta) = cosleft(frac{pi}{2} - theta right) = frac{1}{csc(theta)}


The reciprocal of sine is cosecant, i.e., the reciprocal of {{math|sin(A)}} is {{math|csc(A)}}, or cosec(A). Cosecant gives the ratio of the length of the hypotenuse to the length of the opposite side:
csc(A) = frac{1}{sin(A)} = frac {textrm{hypotenuse}} {textrm{opposite}} = frac{h}{a}.


(File:Arcsine.svg|thumb|180px|The usual principal values of the {{math|arcsin(x)}} function graphed on the cartesian plane. Arcsin is the inverse of sin.)The inverse function of sine is arcsine (arcsin or asin) or inverse sine ({{math|sin{{sup|-1}}}}). As sine is non-injective, it is not an exact inverse function but a partial inverse function. For example, {{math|sin(0) {{=}} 0}}, but also {{math|sin({{pi}}) {{=}} 0}}, {{math|sin(2{{pi}}) {{=}} 0}} etc. It follows that the arcsine function is multivalued: {{math|arcsin(0) {{=}} 0}}, but also {{math|arcsin(0) {{=}} {{pi}}}}, {{math|arcsin(0) {{=}} 2{{pi}}}}, etc. When only one value is desired, the function may be restricted to its principal branch. With this restriction, for each x in the domain the expression {{math|arcsin(x)}} will evaluate only to a single value, called its principal value.
theta = arcsin left( frac{text{opposite}}{text{hypotenuse}} right) = sin^{-1} left( frac {a}{h} right).
k is some integer:
sin(y) = x iff & y = arcsin(x) + 2pi k , text{ or }
& y = pi - arcsin(x) + 2pi k
end{align}Or in one equation:
sin(y) = x iff y = (-1)^k arcsin(x) + pi k
Arcsin satisfies:
sin(arcsin(x)) = x!
arcsin(sin(theta)) = thetaquadtext{for }-frac{pi}{2} leq theta leq frac{pi}{2}.


{{See also|List of integrals of trigonometric functions|Differentiation of trigonometric functions}}For the sine function:
f(x) = sin(x)
The derivative is:
f'(x) = cos(x)
The antiderivative is:
int f(x),dx = -cos(x) + C
C denotes the constant of integration.

Other trigonometric functions

(File:Sine cosine one period.svg|right|thumb|The sine and cosine functions are related in multiple ways. The two functions are out of phase by 90°: sin(pi/2 - x) = cos(x) for all angles x. Also, the derivative of the function {{math|sin(x)}} is {{math|cos(x)}}.)It is possible to express any trigonometric function in terms of any other (up to a plus or minus sign, or using the sign function).Sine in terms of the other common trigonometric functions:{| class="wikitable"!colspan="1" rowspan="3"|!colspan="1" rowspan="3"|f θ!colspan="5"|Using plus/minus (±)!colspan="1"|Using sign function (sgn)!colspan="1" rowspan=2|f θ =!colspan="4"|± per Quadrant!rowspan="2"|f θ =! I! II! III! IV!rowspan="2"|cos|sin(theta)|= pmsqrt{1 - cos^2(theta)}| +| +| −| −|= sgnleft( cos left(theta - frac{pi}{2}right)right) sqrt{1 - cos^2(theta)}|cos(theta)|= pmsqrt{1 - sin^2(theta)}| +| −| −| +|= sgnleft( sin left(theta+ frac{pi}{2}right)right) sqrt{1 - sin^2(theta)}!rowspan="2"|cot| sin(theta)| = pmfrac{1}{sqrt{1 + cot^2(theta)}}| +| +| −| −| = sgnleft( cotleft( frac{theta}{2}right)right) frac{1}{sqrt{1 + cot^2(theta)}}|cot(theta)| = pmfrac{sqrt{1 - sin^2(theta)}}{sin(theta)}| +| −| −| +| = sgnleft( sin left(theta+ frac{pi}{2}right)right) frac{sqrt{1 - sin^2(theta)}}{sin(theta)}!rowspan="2"|tan| sin(theta)| = pmfrac{tan(theta)}{sqrt{1 + tan^2(theta)}} | +| −| −| +| = sgnleft( tanleft(frac{2theta + pi}{4}right)right) frac{tan(theta)}{sqrt{1 + tan^2(theta)}}| tan(theta)| = pmfrac{sin(theta)}{sqrt{1 - sin^2(theta)}}| +| −| −| +| = sgnleft( sin left(theta+ frac{pi}{2}right)right) frac{sin(theta)}{sqrt{1 - sin^2(theta)}}!rowspan="2"|sec| sin(theta)| = pmfrac{sqrt{sec^2(theta) - 1}}{sec(theta)} | +| −| +| −| = sgnleft( sec left( frac{4 theta - pi}{2}right)right) frac{sqrt{sec^2(theta) - 1}}{sec(theta)} | sec(theta)| = pmfrac{1}{sqrt{1 - sin^2(theta)}}| +| −| −| +| = sgnleft( sin left(theta+ frac{pi}{2}right)right) frac{1}{sqrt{1 - sin^2(theta)}}Note that for all equations which use plus/minus (±), the result is positive for angles in the first quadrant.The basic relationship between the sine and the cosine can also be expressed as the Pythagorean trigonometric identity:
cos^2(theta) + sin^2(theta) = 1!
where sin2(x) means (sin(x))2.

Properties relating to the quadrants

(File:Quadrants 01 Pengo.svg|thumb|200px|The four quadrants of a Cartesian coordinate system.)The table below displays many of the key properties of the sine function (sign, monotonicity, convexity), arranged by the quadrant of the argument. For arguments outside those in the table, one may compute the corresponding information by using the periodicity sin(alpha + 360^circ) = sin(alpha) of the sine function.{| class="wikitable" style="text-align: center"!Quadrant!Degrees!Radians!Value!Sign!Monotony!Convexity|1st Quadrant|0^circ

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