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| | Nice to meet you, my title is Refugia. Minnesota is where he's been living for years. Managing people is his profession. Doing ceramics is what her family and her enjoy.<br><br>Check out my web site; std home test ([http://facehack.ir/index.php?do=/profile-110/info/ Recommended Web-site]) |
| {{Function
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| |name = Sine function
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| |image = Sine_one_period.svg
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| |heading1 = 1
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| |parity = odd
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| |domain = (−∞,∞)<ref group=note name=real>For real (not complex) numbers.</ref>
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| |codomain = [−1,1]<ref group=note name=real />
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| |period = 2''π''
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| |heading2 = 1
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| |zero = 0
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| |plusinf =
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| |minusinf =
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| |max = ((2''k'' + ½)''π'', 1) <ref group=note name=intk>Variable ''k'' is an [[integer]]. </ref>
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| |min = ((2''k'' − ½)''π'', −1)
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| |vr1 =
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| |f1 =
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| |vr2 =
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| |f2 =
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| |vr3 =
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| |f3 =
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| |vr4 =
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| |f4 =
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| |vr5 =
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| |f5 =
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| |heading3 = 1
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| |asymptote =
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| |root = ''kπ''
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| |critical = ''kπ'' − π/2
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| |inflection = ''kπ''
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| |fixed = 0
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| |notes = {{reflist|group=note}}
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| }}
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| [[File:Trigono sine en2.svg|right|thumb|<math>\sin \alpha = \frac {\textrm{opposite}} {\textrm{hypotenuse}} </math><br/>For the angle α, the sine function gives the ratio of the length of the opposite side to the length of the hypotenuse.]]
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| [[Image:Sine.svg|right|thumb|The sine function graphed on the Cartesian plane. In this graph, the angle ''x'' is given in [[radian]]s (π = 180°).]]
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| [[File:Sine cosine one period.svg|right|thumb|The sine and cosine functions are related in multiple ways. The derivative of <math>\sin(x)</math> is <math>\cos(x)</math>. Also they are out of phase by 90°: <math>\sin(\pi/2 - x)</math> = <math>\cos(x)</math>. And for a given angle, cos and sin give the respective x, y coordinates on a unit circle.]]
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| In [[mathematics]], the '''sine function''' is a [[trigonometric function]] of an [[angle]]. The sine of an 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 (divided by) the length of the [[hypotenuse|longest side of the triangle]] (i.e. the hypotenuse).
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| Trigonometric functions are commonly defined as [[ratio]]s of two sides of a right triangle containing the angle, and can equivalently be defined as the lengths of various line segments from a [[unit circle]]. More modern definitions express them as [[Series (mathematics)|infinite series]] or as solutions of certain [[differential equation]]s, allowing their extension to arbitrary positive and negative values and even to [[complex number]]s.
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| The sine function is commonly used to model [[periodic function|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.
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| The function sine can be traced to the [[Jyā, koti-jyā and utkrama-jyā|''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.<ref name="Boyer, Carl B. 1991 p. 210">Boyer, Carl B. (1991). A History of Mathematics (Second ed.). John Wiley & Sons, Inc.. ISBN 0-471-54397-7, p. 210.</ref> The word "sine" comes from a [[Latin]] mistranslation of the Arabic ''jiba'', which is a transliteration of the Sanskrit word for half the chord, ''jya-ardha''.<ref>Victor J Katx, A history of mathematics, p210, sidebar 6.1.</ref>
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| == Right-angled triangle definition ==
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| For "opposite" and the "adjacent" side ('''[[Trigonometric functions#Sine.2C_cosine_and_tangent|tangent]]'''), etc.
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| To define the trigonometric functions for an acute angle ''A'', start with any [[right triangle]] that contains the angle ''A''. The three sides of the triangle are named as follows:
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| * The ''adjacent side'' is the side that is in contact with (adjacent to) both the angle we are interested in (angle ''A'') and the right angle, in this case side '''b'''.
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| * 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.
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| * The ''opposite side'' is the side opposite to the angle we are interested in (angle ''A''), in this case side '''a'''.
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| In ordinary [[Euclidean geometry]], according to the [[triangle postulate]] the inside angles of every triangle total 180[[degree (angle)|°]] (π [[radian]]s). Therefore, in a right-angled triangle, the two non-right angles total 90° (π/2 radians), so each of these angles must be greater than 0° and less than 90°. The following definition applies to such angles.
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| The angle ''A'' (having measure α) is the angle between the hypotenuse and the adjacent side.
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| The '''sine''' of an angle is the ratio of the length of the opposite [[Cathetus|side]] to the length of the hypotenuse. In our case
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| does not depend on the size of the particular right triangle chosen, as long as it contains the angle ''A'', since all such triangles are [[similarity (geometry)|similar]].
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| == Relation to slope ==
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| {{Main|Slope}}
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| The trigonometric functions can be defined in terms of the ''rise'', ''run'', and ''[[slope]]'' of a line segment relative to some horizontal line.
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| *When the length of the line segment is 1, sine takes an angle and tells the ''rise''
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| *Sine takes an angle and tells the ''rise'' per unit length of the line segment.
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| *''Rise'' is equal to sin ''θ'' multiplied by the length of the line segment
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| In contrast, cosine is used for the telling the ''run'' from the angle; and tangent is used for telling the ''slope'' from the angle. Arctan is used for telling the angle from the ''slope''.
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| The line segment is the equivalent of the hypotenuse in the right-triangle, and when it has a length of 1 it is also equivalent to the radius of the [[unit circle]].
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| == Relation to the unit circle ==
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| In [[trigonometry]], a unit circle is the circle of radius one centered at the origin (0, 0) in the [[Cartesian coordinate system]].
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| 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 cos ''θ'' and sin ''θ'', respectively. The point's distance from the origin is always 1.
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| Unlike the definitions with the right or left triangle or slope, the angle can be extended to the full set of real arguments by using the [[Unit_circle#Trigonometric_functions_on_the_unit_circle|unit circle]]. This can also be achieved by requiring certain symmetries and that sine be a [[periodic function]].
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| {|
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| |valign="top"| [[File:Unit circle3.png|right|thumb|The unit circle.]]
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| |valign="top"| [[File:Unit circle.svg|Unit circle|right|thumb|Illustration of a unit circle. The radius has a length of 1. The variable ''t'' is an [[angle]] measure.]]
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| |valign="top"| [[File:Trig functions on unit circle.PNG|thumb|Point ''P''(''x'',''y'') on the circle of unit radius at an [[obtuse angle]] θ > π/2]]
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| |}
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| [[File:Sine curve drawing animation.gif|frame|left|Animation showing the graphing process of ''y'' = sin ''x'' (where ''x'' is the angle in radians) using a unit circle. The blue arc around the unit circle (in green) and the blue line at right have the same length, equal to the angle in radians.]]
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| {{clr}}
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| == Identities ==
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| {{See also|List of trigonometric identities}}
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| Exact identities (using [[radian]]s):
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| These apply for all values of <math>\theta</math>.
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| :<math>
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| \begin{align}
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| \sin \theta & = \cos \left(\frac{\pi}{2} - \theta \right) \\
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| & = \frac{1}{\csc \theta}
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| \end{align}
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| </math>
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| === Reciprocal ===
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| The [[multiplicative inverse|reciprocal]] of sine is cosecant, i.e. the reciprocal of sin(''A'') is csc(''A''), or cosec(''A''). Cosecant gives the ratio of the length of the hypotenuse to the length of the opposite side:
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| :<math>\csc A = \frac {1}{\sin A} = \frac {\textrm{hypotenuse}} {\textrm{opposite}} = \frac {h} {a}. </math>
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| === Inverse ===
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| [[File:Arcsine.svg|thumb|180px|The usual principal values of the arcsin(x) function graphed on the cartesian plane. Arcsin is the inverse of sin.]]
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| The [[inverse function]] of sine is arcsine (arcsin or asin) or inverse sine (sin{{sup|−1}}). As sine is non-[[injective]], it is not an exact inverse function but a partial inverse function. For example, sin(0) = 0, but also sin(π) = 0, sin(2π) = 0 etc. It follows that the arcsine function is multivalued: arcsin(0) = 0, but also arcsin(0) = π, arcsin(0) = 2π, 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 arcsin(''x'') will evaluate only to a single value, called its [[principal value]].
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| :<math>\theta = \arcsin \left( \frac{\text{opposite}}{\text{hypotenuse}} \right) = \sin^{-1} \left( \frac {a} {h} \right).</math>
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| ''k'' is some integer:
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| :<math>\begin{align}
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| \sin y = x \ \Leftrightarrow\ & y = \arcsin x + 2k\pi , \text{ or }\\
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| & y = \pi - \arcsin x + 2k\pi
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| \end{align}</math>
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| Or in one equation:
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| :<math>\sin y = x \ \Leftrightarrow\ y = (-1)^k \arcsin x + k\pi </math>
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| Arcsin satisfies:
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| :<math>\sin(\arcsin x) = x\!</math>
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| and
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| :<math>\arcsin(\sin \theta) = \theta\quad\text{for }-\pi/2 \leq \theta \leq \pi/2.</math>
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| === Calculus ===
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| {{See also|List of integrals of trigonometric functions|Differentiation of trigonometric functions}}
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| For the sine function:
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| :<math> f(x) = \sin x \,</math>
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| The derivative is:
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| :<math> f'(x) = \cos x \,</math>
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| The antiderivative is:
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| :<math>\int f(x)\,dx = -\cos x + C </math>
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| ''C'' denotes the [[constant of integration]].
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| === Other trigonometric functions ===
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| [[File:Quadrants 01 Pengo.svg|thumb|350px|The four quadrants of a Cartesian coordinate system.]]
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| 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]]).
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| Sine in terms of the other common [[trigonometric functions]]:
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| {| class="wikitable"
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| |-
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| !colspan="1" rowspan="3"|
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| !colspan="1" rowspan="3"|f ''θ''
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| !colspan="5"|Using plus/minus (±)
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| !colspan="1"|Using sign function (sgn)
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| |-
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| !colspan="1" rowspan=2|f ''θ'' =
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| !colspan="4"|± per Quadrant
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| !rowspan="2"|f ''θ'' =
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| |-
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| ! I
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| ! II
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| ! III
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| ! IV
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| |-
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| !rowspan="2"|cos
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| |<math>\sin \theta </math>
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| |<math>= \pm\sqrt{1 - \cos^2 \theta}</math>
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| | +
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| | +
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| | -
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| | -
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| |<math>= \sgn( \cos (\theta - \frac{\pi}{2})) \sqrt{1 - \cos^2\theta}</math>
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| |-
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| |<math>\cos \theta </math>
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| |<math>= \pm\sqrt{1 - \sin^2\theta}</math>
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| | +
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| | -
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| | -
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| | +
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| |<math>= \sgn( \sin (\theta+ \frac{\pi}{2})) \sqrt{1 - \sin^2\theta}</math>
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| |-
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| !rowspan="2"|cot
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| | <math>\sin \theta </math>
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| | <math>= \pm\frac{1}{\sqrt{1 + \cot^2 \theta}}</math>
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| | +
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| | +
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| | -
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| | -
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| | <math>= \sgn( \cot( \frac{\theta}{2})) \frac{1}{\sqrt{1 + \cot^2 \theta}}</math>
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| |-
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| |<math>\cot \theta</math>
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| |<math> = \pm\frac{\sqrt{1 - \sin^2 \theta}}{\sin \theta}</math>
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| | +
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| | -
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| | -
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| | +
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| | <math>= \sgn( \sin (\theta+ \frac{\pi}{2})) \frac{\sqrt{1 - \sin^2 \theta}}{\sin \theta}</math>
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| |-
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| !rowspan="2"|tan
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| | <math>\sin \theta </math>
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| | <math>= \pm\frac{\tan \theta}{\sqrt{1 + \tan^2 \theta}} </math>
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| | +
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| | -
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| | -
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| | +
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| | <math>= \sgn( \tan(\frac{2\theta + \pi}{4})) \frac{\tan \theta}{\sqrt{1 + \tan^2 \theta}} </math>
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| |-
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| | <math>\tan \theta </math>
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| | <math>= \pm\frac{\sin \theta}{\sqrt{1 - \sin^2 \theta}}</math>
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| | +
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| | -
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| | -
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| | +
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| | <math>= \sgn( \sin (\theta+ \frac{\pi}{2})) \frac{\sin \theta}{\sqrt{1 - \sin^2 \theta}}</math>
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| |-
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| !rowspan="2"|sec
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| | <math>\sin \theta </math>
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| | <math>= \pm\frac{\sqrt{\sec^2 \theta - 1}}{\sec \theta} </math>
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| | +
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| | -
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| | +
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| | -
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| | <math>= \sgn( \sec ( \frac{4 \theta - \pi}{2})) \frac{\sqrt{\sec^2 \theta - 1}}{\sec \theta} </math>
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| |-
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| | <math>\sec \theta </math>
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| | <math>= \pm\frac{1}{\sqrt{1 - \sin^2 \theta}}</math>
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| | +
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| | -
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| | -
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| | +
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| | <math>= \sgn( \sin (\theta+ \frac{\pi}{2})) \frac{1}{\sqrt{1 - \sin^2 \theta}}</math>
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| |}
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| Note that for all equations which use plus/minus (±), the result is positive for angles in the first quadrant.
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| The basic relationship between the sine and the cosine can also be expressed as the [[Pythagorean trigonometric identity]]:
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| :<math>\cos^2\theta + \sin^2\theta = 1\!</math>
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| where sin<sup>2</sup>''x'' means (sin(''x''))<sup>2</sup>.
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| == Properties relating to the quadrants ==
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| Over the four quadrants of the sine function is as follows.
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| {| class="wikitable"
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| |-
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| ![[Cartesian_coordinate_system#Quadrants_and_octants|Quadrant]]
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| ![[Degree (angle)|Degrees]]
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| ![[Radian]]s
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| !Value
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| ![[Sign (mathematics)|Sign]]
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| ![[Monotonic function|Monotony]]
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| ![[Convex function|Convexity]]
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| |-
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| |1st Quadrant
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| |<math>0^\circ<x<90^\circ</math>
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| |<math>0<x< \pi/2 </math>
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| |<math>0<\sin x<1</math>
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| |<math>+</math>
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| |increasing
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| |concave
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| |-
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| |2nd Quadrant
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| |<math>90^\circ<x<180^\circ</math>
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| |<math>\pi/2<x<\pi</math>
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| |<math>0<\sin x<1</math>
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| |<math>+</math>
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| |decreasing
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| |concave
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| |-
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| |3rd Quadrant
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| |<math>180^\circ<x<270^\circ</math>
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| |<math>\pi<x<3\pi/2</math>
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| |<math>-1<\sin x<0</math>
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| |<math>-</math>
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| |decreasing
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| |convex
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| |-
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| |4th Quadrant
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| |<math>270^\circ<x<360^\circ</math>
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| |<math>3\pi/2<x<2\pi</math>
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| |<math>-1<\sin x<0</math>
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| |<math>-</math>
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| |increasing
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| |convex
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| |}
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| Points between the quadrants. k is an [[integer]].
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| [[File:Sine quads 01 Pengo.svg|thumb|390px|The quadrants of the unit circle and of ''sin x'', using the [[Cartesian coordinate system]].]]
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| {| class="wikitable"
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| |-
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| ![[Degree (angle)|Degrees]]
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| ![[Radian]]s
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| 0 ≤ ''x'' < 2π
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| !Radians
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| !sin ''x''
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| !Point type
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| |-
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| |0°
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| |0
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| |<math>2 \pi k</math>
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| |0
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| |[[Root of a function|Root]], [[Inflection point|Inflection]]
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| |-
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| |90°
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| |<math>\pi/2</math>
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| |<math>2 \pi k + \pi/2 </math>
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| |1
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| |[[Maxima and minima|Maxima]]
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| |-
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| |180°
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| |<math>\pi</math>
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| |<math>2 \pi k - \pi</math>
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| |0
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| |[[Root of a function|Root]], [[Inflection point|Inflection]]
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| |-
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| |270°
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| |<math>3\pi/2</math>
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| |<math>2 \pi k - \pi/2 </math>
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| | -1
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| |[[Minima]]
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| |}
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| For arguments outside those in the table, get the value using the fact the sine function has a period of 360° (or 2π rad): <math>\sin(\alpha + 360^\circ) = \sin(\alpha)</math>, or use <math>\sin(\alpha + 180^\circ) = -\sin(\alpha)</math>.
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| Or use <math>\cos(x)= \frac{e^{ix}+e^{-ix}}{2} </math> and <math>\sin(x)=\frac{e^{ix}-e^{-ix}}{2i}</math>.
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| For complement of sine, we have <math>\sin(180^\circ-\alpha) = \sin(\alpha)</math>.
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| == Series definition ==
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| [[Image:Taylorsine.svg|thumb|right|The sine function (blue) is closely approximated by its [[Taylor's theorem|Taylor polynomial]] of degree 7 (pink) for a full cycle centered on the origin.]]
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| [[File:Sine GIF.gif|thumb|right|This animation shows how including more and more terms in the partial sum of its Taylor series gradually builds up a sine curve.]]
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| Using only geometry and properties of [[limit of a function|limits]], it can be shown that the [[derivative]] of sine is cosine, and that the derivative of cosine is the negative of sine.
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| Using the reflection from the calculated geometric derivation of the sine is with the 4n + k-th derivative at the point 0:
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| :<math>\sin^{(4n+k)}(0)=\begin{cases}
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| 0 & \text{when } k=0 \\
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| 1 & \text{when } k=1 \\
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| 0 & \text{when } k=2 \\
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| -1 & \text{when } k=3 \end{cases}</math>
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| This gives the following Taylor series expansion at x = 0. One can then use the theory of [[Taylor series]] to show that the following identities hold for all [[real number]]s ''x'' (where x is the angle in radians) :<ref>See Ahlfors, pages 43–44.</ref>
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| :<math>
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| \begin{align}
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| \sin x & = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \\[8pt]
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| & = \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}x^{2n+1} \\[8pt]
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| \end{align}
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| </math>
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| If ''x'' were expressed in degrees then the series would contain messy factors involving powers of π/180: if ''x'' is the number of degrees, the number of radians is ''y'' = π''x'' /180, so
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| :<math>\begin{align}
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| \sin x_\mathrm{deg} & = \sin y_\mathrm{rad} \\
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| & = \frac{\pi}{180} x - \left (\frac{\pi}{180} \right )^3\ \frac{x^3}{3!} + \left (\frac{\pi}{180} \right )^5\ \frac{x^5}{5!} - \left (\frac{\pi}{180} \right )^7\ \frac{x^7}{7!} + \cdots .
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| \end{align}</math>
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| The series formulas for the sine and [[cosine]] are uniquely determined, up to the choice of unit for angles, by the requirements that
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| :<math>
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| \begin{align}
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| \sin 0 = 0 & \text{ and } \sin{2x} = 2 \sin x \cos x \\
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| \cos^2 x + \sin^2 x = 1 & \text{ and } \cos{2x} = \cos^2 x - \sin^2 x \\
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| \end{align}
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| </math>
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| The radian is the unit that leads to the expansion with leading coefficient 1 for the sine and is determined by the additional requirement that
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| :<math>
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| \sin x \approx x \text{ when } x \approx 0.
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| </math>
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| The coefficients for both the sine and cosine series may therefore be derived by substituting their expansions into the pythagorean and double angle identities, taking the leading coefficient for the sine to be 1, and matching the remaining coefficients.
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| In general, mathematically important relationships between the sine and cosine functions and the [[exponential function]] (see, for example, [[Euler's formula]]) are substantially simplified when angles are expressed in radians, rather than in degrees, grads or other units. Therefore, in most branches of mathematics beyond practical geometry, angles are generally assumed to be expressed in radians.
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| A similar series is [[Gregory's series]] for [[arctan]], which is obtained by omitting the factorials in the denominator.
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| ==Continued fraction==
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| The sine function can also be represented as a [[generalized continued fraction]]:
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| :<math> \sin x =
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| \cfrac{x}{1 + \cfrac{x^2}{2\cdot3-x^2 +
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| \cfrac{2\cdot3 x^2}{4\cdot5-x^2 +
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| \cfrac{4\cdot5 x^2}{6\cdot7-x^2 + \ddots}}}}.
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| </math>
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| The continued fraction representation expresses the [[real number]] values, both [[rational number|rational]] and [[irrational number|irrational]], of the sine function.
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| ==Fixed point==
| |
| [[Image:Sine fixed point.svg|thumb|The fixed point iteration ''x''<sub>''n''+1</sub> = sin ''x''<sub>''n''</sub> with initial value ''x''<sub>0</sup> = 2 converges to 0.]]
| |
| Zero is the only real [[Fixed point (mathematics)|fixed point]] of the sine function; in other words the only intersection of the sine function and the [[identity function]] is sin(0) = 0.
| |
| {{clear}}
| |
| | |
| ==Arc length==
| |
| Strict formulae: <math> \int_a^b{\sqrt{1+\cos(x)^2}} dx </math>
| |
| | |
| Imprecise formulae for sine curve length from 0 to ''x'': <math> \frac{45}{37} x + \frac{3}{29} \sin(2 x) - \frac{x}{4796} </math>
| |
| <ref>[http://sadler.su/gp/sine.html Genetic programming: imprecise length of sin(x) curve]</ref>
| |
| | |
| ==Law of sines==
| |
| {{Main|Law of sines}}
| |
| The [[law of sines]] states that for an arbitrary [[triangle]] with sides ''a'', ''b'', and ''c'' and angles opposite those sides ''A'', ''B'' and ''C'':
| |
| | |
| :<math>\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}.</math>
| |
| | |
| This is equivalent to the equality of the first three expressions below:
| |
| | |
| :<math>\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R,</math>
| |
| | |
| where ''R'' is the triangle's [[circumscribed circle|circumradius]].
| |
| | |
| It can be proven by dividing the triangle into two right ones and using the above definition of sine. The law of sines is useful for computing the lengths of the unknown sides in a triangle if two angles and one side are known. This is a common situation occurring in ''[[triangulation]]'', a technique to determine unknown distances by measuring two angles and an accessible enclosed distance.
| |
| | |
| == Values ==
| |
| {{See also|Exact trigonometric constants}}
| |
| [[File:Sin.svg|thumb|350px|sin(x)]]
| |
| [[Image:Unit circle angles.svg|right|thumb|350px|Some common angles (''θ'') shown on the [[unit circle]]. The angles are given in degrees and radians, together with the corresponding intersection point on the unit circle, (cos ''θ'', sin ''θ'').]]
| |
| {| border="1" cellpadding="5" cellspacing="0"
| |
| |-----
| |
| ! style="background:#ffdead;" colspan="3" | ''x'' (angle)
| |
| ! colspan="2" style="background:#ffdead;" | sin ''x''
| |
| |-----
| |
| | style="background:#efefef;" align="Center" | Degrees
| |
| | style="background:#efefef;" | Radians
| |
| | style="background:#efefef;" align="Center" | Grads
| |
| | style="background:#efefef;" align="Center" | Exact
| |
| | style="background:#efefef;" | Decimal
| |
| |-----
| |
| | align="Center" | 0° || align="Center" | 0
| |
| | align="Center" | 0<sup>g</sup>
| |
| | align="Center" rowspan="2" | 0 || rowspan="2" | 0
| |
| |-----
| |
| | align="Center" | 180° || align="Center" | <math>\pi</math>
| |
| | align="Center" | 200<sup>g</sup>
| |
| |-----
| |
| | align="Center" | 15°
| |
| | align="Center" | <math>\frac{\pi}{12}</math>
| |
| | align="Center" | 16{{frac|2|3}}<sup>g</sup>
| |
| | align="Center" rowspan="2" | <math>\frac{\sqrt{6}-\sqrt{2}}{4}</math>
| |
| | rowspan="2" | 0.258819045102521
| |
| |-----
| |
| | align="Center" | 165°
| |
| | align="Center" | <math>\frac{11 \cdot \pi}{12}</math>
| |
| | align="Center" | 183{{frac|1|3}}<sup>g</sup>
| |
| |-----
| |
| | align="Center" | 30°
| |
| | align="Center" | <math>\frac{\pi}{6}</math>
| |
| | align="Center" | 33{{frac|1|3}}<sup>g</sup>
| |
| | align="Center" rowspan="2" | <math>\frac{1}{2}</math>
| |
| | rowspan="2" | 0.5
| |
| |-----
| |
| | align="Center" | 150°
| |
| | align="Center" | <math>\frac{5 \cdot \pi}{6}</math>
| |
| | align="Center" | 166{{frac|2|3}}<sup>g</sup>
| |
| |-----
| |
| | align="Center" | 45°
| |
| | align="Center" | <math>\frac{\pi}{4}</math>
| |
| | align="Center" | 50<sup>g</sup>
| |
| | align="Center" rowspan="2" | <math>\sqrt{\frac{1}{2}}</math>
| |
| | rowspan="2" | 0.707106781186548
| |
| |-----
| |
| | align="Center" | 135°
| |
| | align="Center" | <math>\frac{3 \cdot \pi}{4}</math>
| |
| | align="Center" | 150<sup>g</sup>
| |
| |-----
| |
| | align="Center" | 60°
| |
| | align="Center" | <math>\frac{\pi}{3}</math>
| |
| | align="Center" | 66{{frac|2|3}}<sup>g</sup>
| |
| | align="Center" rowspan="2" | <math>\frac{\sqrt{3}}{2}</math>
| |
| | rowspan="2" | 0.866025403784439
| |
| |-----
| |
| | align="Center" | 120°
| |
| | align="Center" | <math>\frac{2 \cdot \pi}{3}</math>
| |
| | align="Center" | 133{{frac|1|3}}<sup>g</sup>
| |
| |-----
| |
| | align="Center" | 75°
| |
| | align="Center" | <math>\frac{5 \cdot \pi}{12}</math>
| |
| | align="Center" | 83{{frac|1|3}}<sup>g</sup>
| |
| | align="Center" rowspan="2" | <math>\frac{\sqrt{6}+\sqrt{2}}{4}</math>
| |
| | rowspan="2" | 0.965925826289068
| |
| |-----
| |
| | align="Center" | 105°
| |
| | align="Center" | <math>\frac{7 \cdot \pi}{12}</math>
| |
| | align="Center" | 116{{frac|2|3}}<sup>g</sup>
| |
| |-----
| |
| | align="Center" | 90°
| |
| | align="Center" | <math>\frac{\pi}{2}</math>
| |
| | align="Center" | 100<sup>g</sup> || align="Center" | 1
| |
| | 1
| |
| |}
| |
| | |
| A memory aid (note it does not include 15° and 75°):
| |
| | |
| {| border="1" cellpadding="5" cellspacing="0"
| |
| |- bgcolor="white" align="center"
| |
| |''x'' in degrees||0°||30°||45°||60°||90°
| |
| |- bgcolor="white" align="center"
| |
| |''x'' in radians||0||π/6||π/4||π/3||π/2
| |
| |- bgcolor="white" align="center"
| |
| |<math>\mathrm{sin} \, x</math>||<math>\frac {\sqrt 0} 2</math>||<math>\frac {\sqrt 1} 2</math>||<math>\frac {\sqrt 2} 2</math>||<math>\frac {\sqrt 3} 2</math>||<math>\frac {\sqrt 4} 2</math>
| |
| |}
| |
| | |
| 90 degree increments:
| |
| | |
| {| border="1" cellpadding="5" cellspacing="0"
| |
| |- bgcolor="white" align="center"
| |
| |''x'' in degrees||0°||90°||180°||270°||360°
| |
| |- bgcolor="white" align="center"
| |
| |''x'' in radians||0||π/2||π||3π/2||2π
| |
| |- bgcolor="white" align="center"
| |
| |<math>\mathrm{sin} \, x</math>||0||1||0||-1||0
| |
| |}
| |
| | |
| Other values not listed above:
| |
| : <math>\sin\frac{\pi}{60}=\sin 3^\circ=\tfrac{1}{16} \left[2(1-\sqrt3)\sqrt{5+\sqrt5}+\sqrt2(\sqrt5-1)(\sqrt3+1)\right]\,</math> {{OEIS2C|A019812}}
| |
| : <math>\sin\frac{\pi}{30}=\sin 6^\circ=\tfrac{1}{8} \left[\sqrt{6(5-\sqrt5)}-\sqrt5-1\right]\,</math> {{OEIS2C|A019815}}
| |
| : <math>\sin\frac{\pi}{20}=\sin 9^\circ=\tfrac{1}{8} \left[\sqrt2(\sqrt5+1)-2\sqrt{5-\sqrt5}\right]\,</math> {{OEIS2C|A019818}}
| |
| : <math>\sin\frac{\pi}{15}=\sin 12^\circ=\tfrac{1}{8} \left[\sqrt{2(5+\sqrt5)}-\sqrt3(\sqrt5-1)\right]\,</math> {{OEIS2C|A019821}}
| |
| : <math>\sin\frac{\pi}{10}=\sin 18^\circ=\tfrac{1}{4}\left(\sqrt5-1\right)=\tfrac{1}{2}\varphi^{-1}\,</math> {{OEIS2C|A019827}}
| |
| : <math>\sin\frac{7\pi}{60}=\sin 21^\circ=\tfrac{1}{16}\left[2(\sqrt3+1)\sqrt{5-\sqrt5}-\sqrt2(\sqrt3-1)(1+\sqrt5)\right]\,</math> {{OEIS2C|A019830}}
| |
| : <math>\sin\frac{\pi}{8}=\sin 22.5^\circ=\tfrac{1}{2}(\sqrt{2-\sqrt{2}}),</math>
| |
| : <math>\sin\frac{2\pi}{15}=\sin 24^\circ=\tfrac{1}{8}\left[\sqrt3(\sqrt5+1)-\sqrt2\sqrt{5-\sqrt5}\right]\,</math> {{OEIS2C|A019833}}
| |
| : <math>\sin\frac{3\pi}{20}=\sin 27^\circ=\tfrac{1}{8}\left[2\sqrt{5+\sqrt5}-\sqrt2\;(\sqrt5-1)\right]\,</math> {{OEIS2C|A019836}}
| |
| : <math>\sin\frac{11\pi}{60}=\sin 33^\circ=\tfrac{1}{16}\left[2(\sqrt3-1)\sqrt{5+\sqrt5}+\sqrt2(1+\sqrt3)(\sqrt5-1)\right]\,</math> {{OEIS2C|A019842}}
| |
| : <math>\sin\frac{\pi}{5}=\sin 36^\circ=\tfrac14[\sqrt{2(5-\sqrt5)}]\,</math> {{OEIS2C|A019845}}
| |
| : <math>\sin\frac{13\pi}{60}=\sin 39^\circ=\tfrac1{16}[2(1-\sqrt3)\sqrt{5-\sqrt5}+\sqrt2(\sqrt3+1)(\sqrt5+1)]\,</math> {{OEIS2C|A019848}}
| |
| : <math>\sin\frac{7\pi}{30}=\sin 42^\circ=\frac{\sqrt6\sqrt{5+\sqrt5}-\sqrt5+1}{8}\,</math> {{OEIS2C|A019851}}
| |
| | |
| For angles greater than 2π or less than −2π, simply continue to rotate around the circle; sine [[periodic function]] with period 2π:
| |
| | |
| : <math>\sin\theta = \sin\left(\theta + 2\pi k \right),\,</math>
| |
| | |
| for any angle θ and any [[integer]] ''k''.
| |
| | |
| The primitive period (the ''smallest'' positive period) of sine is a full circle, i.e. 2π radians or 360 degrees.
| |
| | |
| ==Relationship to complex numbers==
| |
| {{Main|Trigonometric functions#Relationship to exponential function and complex numbers}}
| |
| [[File:Complex picture.svg|thumb|An illustration of the [[complex plane]]. The [[imaginary number]]s are on the vertical coordinate axis.]]
| |
| Sine is used to determine the [[imaginary part]] of a [[complex number]] given in [[polar coordinates]] (r,φ):
| |
| | |
| :<math> z = r(\cos \varphi + i\sin \varphi )\,</math>
| |
| | |
| the imaginary part is:
| |
| | |
| :<math>\operatorname{Im}(z) = r \sin \varphi</math>
| |
| | |
| r and φ represent the magnitude and angle of the complex number respectively. ''i'' is the [[imaginary unit]]. ''z'' is a [[complex number]].
| |
| | |
| Although dealing with complex numbers, sine's parameter in this usage is still a [[real number]]. Sine can also take a complex number as an argument.
| |
| {{clr}}
| |
| | |
| ===Sine with a complex argument===
| |
| [[File:ComplexPlot-Sin-z-,1024-.jpg|thumb|<math>\sin z\,</math><br/><br/> [[Domain coloring]] of sin(z) over (-π,π) on x and y axes. Brightness indicates absolute magnitude, saturation represents imaginary and real magnitude.]]
| |
| [[File:Sin z vector field 02 Pengo.svg|thumb|sin(z) as a vector field]]
| |
| | |
| The definition of the sine function for complex arguments ''z'':
| |
| | |
| : <math>\begin{align}
| |
| \sin z & = \sum_{n=0}^\infty \frac{(-1)^{n}}{(2n+1)!}z^{2n+1} \\
| |
| & = \frac{e^{i z} - e^{-i z}}{2i}\, \\
| |
| & = \frac{\sinh \left( i z\right) }{i}
| |
| \end{align}</math>
| |
| | |
| where ''i''<sup> 2</sup> = −1, and sinh is [[hyperbolic function|hyperbolic sine]]. This is an [[entire function]]. Also, for purely real ''x'',
| |
| | |
| : <math>\sin x = \operatorname{Im}(e^{i x}). \,</math>
| |
| | |
| For purely imaginary numbers:
| |
| :<math> \sin iy = i \sinh y. \,</math>
| |
| | |
| It is also sometimes useful to express the complex sine function in terms of the real and imaginary parts of its argument:
| |
| | |
| : <math>\begin{align}
| |
| \sin (x + iy) &= \sin x \cos iy + \cos x \sin iy \\
| |
| &= \sin x \cosh y + i \cos x \sinh y.
| |
| \end{align}</math>
| |
| | |
| ==== Partial fraction and product expansions of complex sine ====
| |
| Using the partial fraction expansion technique in [[Complex Analysis]], one can find that the infinite series
| |
| | |
| : <math>\begin{align}
| |
| \sum_{n = -\infty}^{\infty}\frac{(-1)^n}{z-n} = \frac{1}{z} + \sum_{n = 1}^{\infty}\frac{2z}{n^2-z^2}
| |
| \end{align}</math>
| |
| | |
| both converge and are equal to <math>\frac{\pi}{\sin \pi z}</math>.
| |
| | |
| Similarly we can find
| |
| | |
| : <math>\begin{align}
| |
| \frac{\pi^2}{\sin^2 \pi z} = \sum_{-\infty}^\infty \frac{1}{(z-n)^2}.
| |
| \end{align}</math>
| |
| | |
| Using product expansion technique, one can derive
| |
| | |
| : <math>\begin{align}
| |
| \sin \pi z = \pi z \prod_{n = 1}^\infty \Bigl( 1- \frac{z^2}{n^2} \Bigr).
| |
| \end{align}</math>
| |
| | |
| ==== Usage of complex sine ====
| |
| ''sin z'' is found in the [[functional equation]] for the [[Gamma function]],
| |
| :<math>\Gamma(s)\Gamma(1-s)={\pi\over\sin\pi s},</math>
| |
| | |
| which in turn is found in the [[functional equation]] for the [[Riemann zeta-function]],
| |
| :<math>\zeta(s)=2(2\pi)^{s-1}\Gamma(1-s)\sin(\pi s/2)\zeta(1-s).</math>
| |
| | |
| As a [[holomorphic function]], ''sin z'' is a 2D solution of [[Laplace's equation]]:
| |
| :<math>\Delta u(x_1, x_2) = 0.</math>
| |
| | |
| ===Complex graphs===
| |
| | |
| {| style="text-align:center"
| |
| |+ '''Sine function in the complex plane'''
| |
| |[[File:Complex sin real 01 Pengo.svg|1000x130px|none]]
| |
| |[[File:Complex sin imag 01 Pengo.svg|1000x130px|none]]
| |
| |[[File:Complex sin abs 01 Pengo.svg|1000x130px|none]]
| |
| |-
| |
| |real component
| |
| |imaginary component
| |
| |magnitude
| |
| |}
| |
| <br />
| |
| {| style="text-align:center"
| |
| |+ '''Arcsine function in the complex plane'''
| |
| |[[File:Complex arcsin real 01 Pengo.svg|1000x130px|none]]
| |
| |[[File:Complex arcsin imag 01 Pengo.svg|1000x130px|none]]
| |
| |[[File:Complex arcsin abs 01 Pengo.svg|1000x130px|none]]
| |
| |-
| |
| |real component
| |
| |imaginary component
| |
| |magnitude
| |
| |}
| |
| | |
| == History ==
| |
| {{Main|History of trigonometric functions}}
| |
| While the early study of trigonometry can be traced to antiquity, the trigonometric functions as they are in use today were developed in the medieval period.
| |
| The [[Chord (geometry)|chord]] function was discovered by [[Hipparchus]] of [[İznik|Nicaea]] (180–125 BC) and [[Ptolemy]] of [[Egypt (Roman province)|Roman Egypt]] (90–165 AD).
| |
| | |
| The function sine (and cosine) can be traced to the [[Jyā, koti-jyā and utkrama-jyā|''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.<ref name="Boyer, Carl B. 1991 p. 210"/>
| |
| | |
| The first published use of the abbreviations 'sin', 'cos', and 'tan' is by the 16th century French mathematician [[Albert Girard]]; these were further promulgated by Euler (see below). The ''Opus palatinum de triangulis'' of [[Georg Joachim Rheticus]], a student of [[Copernicus]], was probably the first in Europe to define trigonometric functions directly in terms of right triangles instead of circles, with tables for all six trigonometric functions; this work was finished by Rheticus' student Valentin Otho in 1596.
| |
| | |
| In a paper published in 1682, [[Gottfried Leibniz|Leibniz]] proved that sin ''x'' is not an [[algebraic function]] of ''x''.<ref>{{cite book|title=Elements of the History of Mathematics|author=Nicolás Bourbaki|publisher=Springer|year=1994}}</ref> [[Roger Cotes]] computed the derivative of sine in his ''Harmonia Mensurarum'' (1722).<ref>"[http://www.math.usma.edu/people/rickey/hm/CalcNotes/Sine-Deriv.pdf Why the sine has a simple derivative]", in ''[http://www.math.usma.edu/people/rickey/hm/CalcNotes/default.htm Historical Notes for Calculus Teachers]'' by [http://www.math.usma.edu/people/rickey/ V. Frederick Rickey]</ref> [[Leonhard Euler]]'s ''Introductio in analysin infinitorum'' (1748) was mostly responsible for establishing the analytic treatment of trigonometric functions in Europe, also defining them as infinite series and presenting "[[Euler's formula]]", as well as the near-modern abbreviations ''sin., cos., tang., cot., sec.,'' and ''cosec.''<ref name=boyer>See Boyer (1991).</ref>
| |
| | |
| === Etymology ===
| |
| {{Wiktionary|sine}}
| |
| [[Etymology|Etymologically]], the word ''sine'' derives from the [[Sanskrit]] word for chord, ''jiva''*(''jya'' being its more popular synonym). This was [[transliteration|transliterated]] in [[Arabic language|Arabic]] as ''jiba'' جــيــب, abbreviated ''jb'' جــــب . Since Arabic is written without short vowels, "jb" was interpreted as the word ''jaib'' جــيــب, which means "bosom", when the Arabic text was translated in the 12th century into [[Medieval Latin|Latin]] by [[Gerard of Cremona]]. The translator used the Latin equivalent for "bosom", ''[[wikt:sinus|sinus]]'' (which means "bosom" or "bay" or "fold") <ref>See Maor (1998), chapter 3, regarding the etymology.</ref><ref>Victor J Katx, ''A history of mathematics'', p210, sidebar 6.1.</ref> The English form ''sine'' was introduced in the 1590s.
| |
| | |
| == Software implementations ==
| |
| {{See also|Lookup table#Computing sines}}
| |
| The sine function, along with other trigonometric functions, is widely available across programming languages and platforms. In computing, it is typically abbreviated to <code>sin</code>.
| |
| | |
| Some CPU architectures have a built-in instruction for sine, including the Intel x87 FPUs since the 80387.
| |
| | |
| In programming languages, <code>sin</code> is typically either a built-in function or found within the language's standard math library.
| |
| | |
| For example, the [[C standard library]] defines sine functions within [[C mathematical functions|math.h]]: <code>sin([[Double-precision floating-point format|double]])</code>, <code>sinf([[Single-precision floating-point format|float]])</code>, and <code>sinl([[long double]])</code>. The parameter of each is a [[floating point]] value, specifying the angle in radians. Each function returns the same [[data type]] as it accepts. Many other trigonometric functions are also defined in [[C mathematical functions|math.h]], such as for cosine, arc sine, and hyperbolic sine (sinh).
| |
| | |
| Similarly, [[Python (programming language)|Python]], defines <code>math.sin(x)</code> within the built-in <code>math</code> module. Complex sine functions are also available within the <code>cmath</code> module, e.g. <code>cmath.sin(z)</code>. [[CPython]]'s math functions call the [[C (programming language)|C]] <code>math</code> library, and use a [[double-precision floating-point format]].
| |
| | |
| There is no standard algorithm for calculating sine. [[IEEE 754-2008]], the most widely-used standard for floating-point computation, does not address calculating trigonometric functions such as sine.<ref>Grand Challenges of Informatics, Paul Zimmermann. September 20, 2006 – p. 14/31 [http://www.jaist.ac.jp/~bjorner/ae-is-budapest/talks/Sept20pm2_Zimmermann.pdf]</ref> Algorithms for calculating sine may be balanced for such constraints as speed, accuracy, portability, or range of input values accepted. This can lead to different results for different algorithms, especially for special circumstances such as very large inputs, e.g. <code>sin(10{{sup|22}})</code>.
| |
| | |
| A once common programming optimization, used especially in 3D graphics, was to pre-calculate a table of sine values, for example one value per degree. This allowed results to be looked up from a table rather than being calculated in real time. With modern CPU architectures this method may offer no advantage. {{citation needed|date=October 2012}} <!-- Looking up in a table essentially involves copying a number from one memory location to another. How can calculating a sine be as quick as that? A: depends on many things, but mainly if the value in memory is not in cache, then the cache-miss can be relatively expensive, whereas a trig function never needs to touch main memory (or alternative, it can be expensive for the rest of the code to have the CPU cache filled with trig tables, causing cache-misses elsewhere). Probably needs a more definitive answer. Have changed 'typically' to 'may' for now. -->
| |
| | |
| == See also ==
| |
| {{Commons category|Sine function}}
| |
| {{Trigonometry}}
| |
| * [[Aryabhata's sine table]]
| |
| * [[Bhaskara I's sine approximation formula]]
| |
| * [[Discrete sine transform]]
| |
| * [[Euler's formula]]
| |
| * [[Generalized trigonometry]]
| |
| * [[Hyperbolic function]]
| |
| * [[Law of sines]]
| |
| * [[List of periodic functions]]
| |
| * [[List of trigonometric identities]]
| |
| * [[Madhava series]]
| |
| * [[Madhava's sine table]]
| |
| * [[Optical sine theorem]]
| |
| * [[Polar sine]] — a generalization to vertex angles
| |
| * [[Proofs of trigonometric identities]]
| |
| * [[Sine and cosine transforms]]
| |
| * [[Sine quadrant]]
| |
| * [[Sine wave]]
| |
| * [[Sine–Gordon equation]]
| |
| * [[Sinusoidal model]]
| |
| * [[Trigonometric functions]]
| |
| | |
| == References ==
| |
| {{Reflist}}
| |
| | |
| [[Category:Trigonometry]]
| |
| [[Category:Elementary special functions]]
| |
| | |
| [[no:Trigonometriske funksjoner#Sinus, cosinus og tangens]]
| |