Cos 2x 3 Sin X 1 0
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Sine and cosine  

General information  
General definition 
$$ 
Fields of application  Trigonometry, fourier series, etc. 
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
, the sine and cosine functions are denoted simply as
and
.^{[1]}
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 functions
jyā
and
koṭijyā
, used in Indian astronomy during the Gupta period (Aryabhatiya
and
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
, itself a transliteration of the Sanskrit word for half of a chord,
jyaardha
.^{[3]}
The word
cosine
derives from a contraction of the medieval Latin
complementi sinus
.^{[4]}
Notation
[edit]
Sine and cosine are written using functional notation with the abbreviations
sin
and
cos. Often if the argument is simple enough, the function value will be written without parentheses, as
sin
θ
rather than as
sin(θ). Each of sine and cosine is a function of an angle, which is usually expressed in terms of radians or degrees. Except where explicitly stated otherwise, this article assumes that the angle is measured in radians.
Definitions
[edit]
Rightangled triangle definitions
[edit]
To define the sine and cosine of an acute angle
α, start with a right triangle that contains an angle of measure
α; in the accompanying figure, angle
α
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 sidea.  The
hypotenuse
is the side opposite the right angle, in this case sideh. The hypotenuse is always the longest side of a rightangled triangle.  The
adjacent side
is the remaining side, in this case sideb. It forms a side of (and 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:^{[5]}

$$
sin
(
α
)
=
opposite
hypotenuse
cos
(
α
)
=
adjacent
hypotenuse
{\displaystyle \sin(\alpha )={\frac {\textrm {opposite}}{\textrm {hypotenuse}}}\qquad \cos(\alpha )={\frac {\textrm {adjacent}}{\textrm {hypotenuse}}}}
The other trigonometric functions of the angle can be defined similarly; for example, the tangent is the ratio between the opposite and adjacent sides.^{[5]}
As stated, the values
and
appear 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 ratios are the same for each of them.
Unit circle definitions
[edit]
In 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 intersect the unit circle, making an angle of
θ
with the positive half of the
xaxis. The
x– and
ycoordinates of this point of intersection are equal to
cos(θ)
and
sin(θ), respectively. This definition is consistent with the rightangled triangle definition of sine and cosine when 0 <
θ
<
π/2: because the length of the hypotenuse of the unit circle is always 1,
. The length of the opposite side of the triangle is simply the
ycoordinate. A similar argument can be made for the cosine function to show that
when 0 <θ <π/2, even under the new definition using the unit circle.
tan(θ)
is then defined as
, or, equivalently, as the slope of the line segment.
Using the unit circle definition has the advantage that the angle can be extended to any real argument. This can also be achieved by requiring certain symmetries, and that sine be a periodic function.
Complex exponential function definitions
[edit]
The exponential function
is defined on the entire domain of the complex numbers. The definition of sine and cosine can be extended to all complex numbers via

$$
sin
z
=
e
i
z
−
e
−
i
z
2
i
{\displaystyle \sin z={\frac {e^{iz}e^{iz}}{2i}}}

$$
cos
z
=
e
i
z
+
e
−
i
z
2
{\displaystyle \cos z={\frac {e^{iz}+e^{iz}}{2}}}
These can be reversed to give Euler’s formula

$$
e
i
z
=
cos
z
+
i
sin
z
{\displaystyle e^{iz}=\cos z+i\sin z}

$$
e
−
i
z
=
cos
z
−
i
sin
z
{\displaystyle e^{iz}=\cos zi\sin z}
When plotted on the complex plane, the function
for real values of
traces out the unit circle in the complex plane.
When
is a real number sine and cosine simplify to the imaginary and real parts of
or
, as:

$$
sin
x
=
Im
(
e
i
x
)
=
−
Im
(
e
−
i
x
)
{\displaystyle \sin x=\operatorname {Im} (e^{ix})=\operatorname {Im} (e^{ix})}

$$
cos
x
=
Re
(
e
i
x
)
=
Re
(
e
−
i
x
)
{\displaystyle \cos x=\operatorname {Re} (e^{ix})=\operatorname {Re} (e^{ix})}
When
for real values
and
, sine and cosine can be expressed in terms of real sines, cosines, and hyperbolic functions as

$$
sin
z
=
sin
x
cosh
y
+
i
cos
x
sinh
y
cos
z
=
cos
x
cosh
y
−
i
sin
x
sinh
y
{\displaystyle {\begin{aligned}\sin z&=\sin x\cosh y+i\cos x\sinh y\\[5pt]\cos z&=\cos x\cosh yi\sin x\sinh y\end{aligned}}}
Differential equation definition
[edit]
is the solution
to the twodimensional system of differential equations
and
with the initial conditions
and
. One could interpret the unit circle in the above definitions as defining the phase space trajectory of the differential equation with the given initial conditions.
Series definitions
[edit]
The successive derivatives of sine, evaluated at zero, can be used to determine its Taylor series. Using only geometry and properties of limits, it can be shown that the derivative of sine is cosine, and that the derivative of cosine is the negative of sine. This means the successive derivatives of sin(x) are cos(x), sin(x), cos(x), sin(x), continuing to repeat those four functions. The (4n+k)th derivative, evaluated at the point 0:

$$
sin
(
4
n
+
k
)
(
)
=
{
when
k
=
1
when
k
=
1
when
k
=
2
−
1
when
k
=
3
{\displaystyle \sin ^{(4n+k)}(0)={\begin{cases}0&{\text{when }}k=0\\1&{\text{when }}k=1\\0&{\text{when }}k=2\\1&{\text{when }}k=3\end{cases}}}
where the superscript represents repeated differentiation. This implies 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 numbers
x
(where x is the angle in radians):^{[6]}

$$
sin
(
x
)
=
x
−
x
3
3
!
+
x
5
5
!
−
x
7
7
!
+
⋯
=
∑
n
=
∞
(
−
1
)
n
(
2
n
+
1
)
!
x
2
n
+
1
{\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]\end{aligned}}}
Taking the derivative of each term gives the Taylor series for cosine:

$$
cos
(
x
)
=
1
−
x
2
2
!
+
x
4
4
!
−
x
6
6
!
+
⋯
=
∑
n
=
∞
(
−
1
)
n
(
2
n
)
!
x
2
n
{\displaystyle {\begin{aligned}\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}}}
Continued fraction definitions
[edit]
The sine function can also be represented as a generalized continued fraction:

$$
sin
(
x
)
=
x
1
+
x
2
2
⋅
3
−
x
2
+
2
⋅
3
x
2
4
⋅
5
−
x
2
+
4
⋅
5
x
2
6
⋅
7
−
x
2
+
⋱
.
{\displaystyle \sin(x)={\cfrac {x}{1+{\cfrac {x^{2}}{2\cdot 3x^{2}+{\cfrac {2\cdot 3x^{2}}{4\cdot 5x^{2}+{\cfrac {4\cdot 5x^{2}}{6\cdot 7x^{2}+\ddots }}}}}}}}.}

$$
cos
(
x
)
=
1
1
+
x
2
1
⋅
2
−
x
2
+
1
⋅
2
x
2
3
⋅
4
−
x
2
+
3
⋅
4
x
2
5
⋅
6
−
x
2
+
⋱
.
{\displaystyle \cos(x)={\cfrac {1}{1+{\cfrac {x^{2}}{1\cdot 2x^{2}+{\cfrac {1\cdot 2x^{2}}{3\cdot 4x^{2}+{\cfrac {3\cdot 4x^{2}}{5\cdot 6x^{2}+\ddots }}}}}}}}.}
The continued fraction representations can be derived from Euler’s continued fraction formula and express the real number values, both rational and irrational, of the sine and cosine functions.
Identities
[edit]
Exact identities (using radians):
These apply for all values of
.

$$
sin
(
θ
)
=
cos
(
π
2
−
θ
)
=
cos
(
θ
−
π
2
)
{\displaystyle \sin(\theta )=\cos \left({\frac {\pi }{2}}\theta \right)=\cos \left(\theta {\frac {\pi }{2}}\right)}

$$
cos
(
θ
)
=
sin
(
π
2
−
θ
)
=
sin
(
θ
+
π
2
)
{\displaystyle \cos(\theta )=\sin \left({\frac {\pi }{2}}\theta \right)=\sin \left(\theta +{\frac {\pi }{2}}\right)}
Reciprocals
[edit]
The 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. Similarly, the reciprocal of cosine is secant, which gives the ratio of the length of the hypotenuse to that of the adjacent side.

$$
csc
(
A
)
=
1
sin
(
A
)
=
hypotenuse
opposite
{\displaystyle \csc(A)={\frac {1}{\sin(A)}}={\frac {\textrm {hypotenuse}}{\textrm {opposite}}}}

$$
sec
(
A
)
=
1
cos
(
A
)
=
hypotenuse
adjacent
{\displaystyle \sec(A)={\frac {1}{\cos(A)}}={\frac {\textrm {hypotenuse}}{\textrm {adjacent}}}}
Inverses
[edit]
The inverse function of sine is arcsine (arcsin or asin) or inverse sine (sin^{−1}
). The inverse function of cosine is arccosine (arccos, acos, or
cos^{−1}
). (The superscript of −1 in
sin^{−1}
and
cos^{−1}
denotes the inverse of a function, not exponentiation.) As sine and cosine are not injective, their inverses are not exact inverse functions, but partial inverse functions. 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. The standard range of principal values for arcsin is from
−π/2
to
π
and the standard range for arccos is from 0 to
π.

$$
θ
=
arcsin
(
opposite
hypotenuse
)
=
arccos
(
adjacent
hypotenuse
)
.
{\displaystyle \theta =\arcsin \left({\frac {\text{opposite}}{\text{hypotenuse}}}\right)=\arccos \left({\frac {\text{adjacent}}{\text{hypotenuse}}}\right).}
where (for some integer
k):

$$
sin
(
y
)
=
x
⟺
y
=
arcsin
(
x
)
+
2
π
k
,
or
y
=
π
−
arcsin
(
x
)
+
2
π
k
cos
(
y
)
=
x
⟺
y
=
arccos
(
x
)
+
2
π
k
,
or
y
=
−
arccos
(
x
)
+
2
π
k
{\displaystyle {\begin{aligned}\sin(y)=x\iff &y=\arcsin(x)+2\pi k,{\text{ or }}\\&y=\pi \arcsin(x)+2\pi k\\\cos(y)=x\iff &y=\arccos(x)+2\pi k,{\text{ or }}\\&y=\arccos(x)+2\pi k\end{aligned}}}
By definition, arcsin and arccos satisfy the equations:

$$
sin
(
arcsin
(
x
)
)
=
x
cos
(
arccos
(
x
)
)
=
x
{\displaystyle \sin(\arcsin(x))=x\qquad \cos(\arccos(x))=x}
and

$$
arcsin
(
sin
(
θ
)
)
=
θ
for
−
π
2
≤
θ
≤
π
2
arccos
(
cos
(
θ
)
)
=
θ
for
≤
θ
≤
π
{\displaystyle {\begin{aligned}\arcsin(\sin(\theta ))=\theta \quad &{\text{for}}\quad {\frac {\pi }{2}}\leq \theta \leq {\frac {\pi }{2}}\\\arccos(\cos(\theta ))=\theta \quad &{\text{for}}\quad 0\leq \theta \leq \pi \end{aligned}}}
Pythagorean trigonometric identity
[edit]
The basic relationship between the sine and the cosine is the Pythagorean trigonometric identity:^{[1]}

$$
cos
2
(
θ
)
+
sin
2
(
θ
)
=
1
{\displaystyle \cos ^{2}(\theta )+\sin ^{2}(\theta )=1}
where sin^{2}(x) means (sin(x))^{2}.
Double angle formulas
[edit]
Sine and cosine satisfy the following double angle formulas:

$$
sin
(
2
θ
)
=
2
sin
(
θ
)
cos
(
θ
)
{\displaystyle \sin(2\theta )=2\sin(\theta )\cos(\theta )}

$$
cos
(
2
θ
)
=
cos
2
(
θ
)
−
sin
2
(
θ
)
=
2
cos
2
(
θ
)
−
1
=
1
−
2
sin
2
(
θ
)
{\displaystyle \cos(2\theta )=\cos ^{2}(\theta )\sin ^{2}(\theta )=2\cos ^{2}(\theta )1=12\sin ^{2}(\theta )}
The cosine double angle formula implies that sin^{2}
and cos^{2}
are, themselves, shifted and scaled sine waves. Specifically,^{[7]}

$$
sin
2
(
θ
)
=
1
−
cos
(
2
θ
)
2
cos
2
(
θ
)
=
1
+
cos
(
2
θ
)
2
{\displaystyle \sin ^{2}(\theta )={\frac {1\cos(2\theta )}{2}}\qquad \cos ^{2}(\theta )={\frac {1+\cos(2\theta )}{2}}}
The graph shows both the sine function and the sine squared function, with the sine in blue and sine squared in red. Both graphs have the same shape, but with different ranges of values, and different periods. Sine squared has only positive values, but twice the number of periods.
Derivative and integrals
[edit]
The derivatives of sine and cosine are:

$$
d
d
x
sin
(
x
)
=
cos
(
x
)
d
d
x
cos
(
x
)
=
−
sin
(
x
)
{\displaystyle {\frac {d}{dx}}\sin(x)=\cos(x)\qquad {\frac {d}{dx}}\cos(x)=\sin(x)}
and their antiderivatives are:

$$
∫
sin
(
x
)
d
x
=
−
cos
(
x
)
+
C
{\displaystyle \int \sin(x)\,dx=\cos(x)+C}

$$
∫
cos
(
x
)
d
x
=
sin
(
x
)
+
C
{\displaystyle \int \cos(x)\,dx=\sin(x)+C}
where
C
denotes the constant of integration.^{[1]}
Properties relating to the quadrants
[edit]
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
of the sine function.
Quadrant  Angle  Sine  Cosine  

Degrees  Radians  Sign  Monotony  Convexity  Sign  Monotony  Convexity  
1st quadrant, I 
$$ 
$$ 
$$ 
increasing  concave 
$$ 
decreasing  concave 
2nd quadrant, II 
$$ 
$$ 
$$ 
decreasing  concave 
$$ 
decreasing  convex 
3rd quadrant, III 
$$ 
$$ 
$$ 
decreasing  convex 
$$ 
increasing  convex 
4th quadrant, IV 
$$ 
$$ 
$$ 
increasing  convex 
$$ 
increasing  concave 
The following table gives basic information at the boundary of the quadrants.
Degrees  Radians 
$$ 
$$ 


Value  Point type  Value  Point type  
$$ 
$$ 
$$ 
Root, inflection 
$$ 
Maximum 
$$ 
$$ 
$$ 
Maximum 
$$ 
Root, inflection 
$$ 
$$ 
$$ 
Root, inflection 
$$ 
Minimum 
$$ 
$$ 
$$ 
Minimum 
$$ 
Root, inflection 
Fixed points
[edit]
Zero is the only real fixed point of the sine function; in other words the only intersection of the sine function and the identity function is
. The only real fixed point of the cosine function is called the Dottie number. That is, the Dottie number is the unique real root of the equation
The decimal expansion of the Dottie number is
.^{[8]}
Arc length
[edit]
The arc length of the sine curve between
and
is

$$
∫
t
1
+
cos
2
(
x
)
d
x
=
2
E
(
t
,
1
/
2
)
,
{\displaystyle \int _{0}^{t}\!{\sqrt {1+\cos ^{2}(x)}}\,dx={\sqrt {2}}\operatorname {E} (t,1/{\sqrt {2}}),}
where
is the incomplete elliptic integral of the second kind with modulus
. It cannot be expressed using elementary functions.
The arc length for a full period is^{[9]}

$$
L
=
4
2
π
3
Γ
(
1
/
4
)
2
+
Γ
(
1
/
4
)
2
2
π
=
7.640395578
…
{\displaystyle L={\frac {4{\sqrt {2\pi ^{3}}}}{\Gamma (1/4)^{2}}}+{\frac {\Gamma (1/4)^{2}}{\sqrt {2\pi }}}=7.640395578\ldots }
where
is the gamma function. This can also be written using
and the lemniscate constant.^{[9]}
^{[10]}
Law of sines
[edit]
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:

$$
sin
A
a
=
sin
B
b
=
sin
C
c
.
{\displaystyle {\frac {\sin A}{a}}={\frac {\sin B}{b}}={\frac {\sin C}{c}}.}
This is equivalent to the equality of the first three expressions below:

$$
a
sin
A
=
b
sin
B
=
c
sin
C
=
2
R
,
{\displaystyle {\frac {a}{\sin A}}={\frac {b}{\sin B}}={\frac {c}{\sin C}}=2R,}
where
R
is the triangle’s circumradius.
It can be proved 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.
Law of cosines
[edit]
The law of cosines states that for an arbitrary triangle with sides
a,
b, and
c
and angles opposite those sides
A,
B
and
C:

$$
a
2
+
b
2
−
2
a
b
cos
(
C
)
=
c
2
{\displaystyle a^{2}+b^{2}2ab\cos(C)=c^{2}}
In the case where
,
and this becomes the Pythagorean theorem: for a right triangle,
where
c
is the hypotenuse.
Special values
[edit]
For certain integral numbers
x
of degrees, the values of sin(x) and cos(x) are particularly simple and can be expressed without nested square roots. A table of these angles is given below. For more complex angle expressions see Exact trigonometric values § Common angles.
Angle, x 
sin(x)  cos(x)  

Degrees  Radians  Gradians  Turns  Exact  Decimal  Exact  Decimal 
0°  0  ^{g}  0  0  0  1  1 
15° 
1 / 12 π 
16+ 2 / 3 ^{g} 
1 / 24 
$$ 
0.2588 
$$ 
0.9659 
30° 
1 / 6 π 
33+ 1 / 3 ^{g} 
1 / 12 
1 / 2 
0.5 
$$ 
0.8660 
45° 
1 / 4 π 
50^{g} 
1 / 8 
$$ 
0.7071 
$$ 
0.7071 
60° 
1 / 3 π 
66+ 2 / 3 ^{g} 
1 / 6 
$$ 
0.8660 
1 / 2 
0.5 
75° 
5 / 12 π 
83+ 1 / 3 ^{g} 
5 / 24 
$$ 
0.9659 
$$ 
0.2588 
90° 
1 / 2 π 
100^{g} 
1 / 4 
1  1  0  0 
90 degree increments:
x in degrees 
0°  90°  180°  270°  360° 

x in radians 
π/2  π  3π/2  2π  
x in gons 
100^{g} 
200^{g} 
300^{g} 
400^{g} 

x in turns 
1/4  1/2  3/4  1  
sin x 
1  −1  0  
cos x 
1  −1  1 
Relationship to complex numbers
[edit]
Sine and cosine are used to connect the real and imaginary parts of a complex number with its polar coordinates (r,
φ):

$$
z
=
r
(
cos
(
φ
)
+
i
sin
(
φ
)
)
{\displaystyle z=r(\cos(\varphi )+i\sin(\varphi ))}
The real and imaginary parts are:

$$
Re
(
z
)
=
r
cos
(
φ
)
{\displaystyle \operatorname {Re} (z)=r\cos(\varphi )}

$$
Im
(
z
)
=
r
sin
(
φ
)
{\displaystyle \operatorname {Im} (z)=r\sin(\varphi )}
where
r
and
φ
represent the magnitude and angle of the complex number
z.
For any real number
θ, Euler’s formula says that:

$$
e
i
θ
=
cos
(
θ
)
+
i
sin
(
θ
)
{\displaystyle e^{i\theta }=\cos(\theta )+i\sin(\theta )}
Therefore, if the polar coordinates of
z
are (r,
φ),
Complex arguments
[edit]
Applying the series definition of the sine and cosine to a complex argument,
z, gives:

$$
sin
(
z
)
=
∑
n
=
∞
(
−
1
)
n
(
2
n
+
1
)
!
z
2
n
+
1
=
e
i
z
−
e
−
i
z
2
i
=
sinh
(
i
z
)
i
=
−
i
sinh
(
i
z
)
cos
(
z
)
=
∑
n
=
∞
(
−
1
)
n
(
2
n
)
!
z
2
n
=
e
i
z
+
e
−
i
z
2
=
cosh
(
i
z
)
{\displaystyle {\begin{aligned}\sin(z)&=\sum _{n=0}^{\infty }{\frac {(1)^{n}}{(2n+1)!}}z^{2n+1}\\&={\frac {e^{iz}e^{iz}}{2i}}\\&={\frac {\sinh \left(iz\right)}{i}}\\&=i\sinh \left(iz\right)\\\cos(z)&=\sum _{n=0}^{\infty }{\frac {(1)^{n}}{(2n)!}}z^{2n}\\&={\frac {e^{iz}+e^{iz}}{2}}\\&=\cosh(iz)\\\end{aligned}}}
where sinh and cosh are the hyperbolic sine and cosine. These are entire functions.
It is also sometimes useful to express the complex sine and cosine functions in terms of the real and imaginary parts of its argument:

$$
sin
(
x
+
i
y
)
=
sin
(
x
)
cos
(
i
y
)
+
cos
(
x
)
sin
(
i
y
)
=
sin
(
x
)
cosh
(
y
)
+
i
cos
(
x
)
sinh
(
y
)
cos
(
x
+
i
y
)
=
cos
(
x
)
cos
(
i
y
)
−
sin
(
x
)
sin
(
i
y
)
=
cos
(
x
)
cosh
(
y
)
−
i
sin
(
x
)
sinh
(
y
)
{\displaystyle {\begin{aligned}\sin(x+iy)&=\sin(x)\cos(iy)+\cos(x)\sin(iy)\\&=\sin(x)\cosh(y)+i\cos(x)\sinh(y)\\\cos(x+iy)&=\cos(x)\cos(iy)\sin(x)\sin(iy)\\&=\cos(x)\cosh(y)i\sin(x)\sinh(y)\\\end{aligned}}}
Partial fraction and product expansions of complex sine
[edit]
Using the partial fraction expansion technique in complex analysis, one can find that the infinite series

$$
∑
n
=
−
∞
∞
(
−
1
)
n
z
−
n
=
1
z
−
2
z
∑
n
=
1
∞
(
−
1
)
n
n
2
−
z
2
{\displaystyle \sum _{n=\infty }^{\infty }{\frac {(1)^{n}}{zn}}={\frac {1}{z}}2z\sum _{n=1}^{\infty }{\frac {(1)^{n}}{n^{2}z^{2}}}}
both converge and are equal to
. Similarly, one can show that

$$
π
2
sin
2
(
π
z
)
=
∑
n
=
−
∞
∞
1
(
z
−
n
)
2
.
{\displaystyle {\frac {\pi ^{2}}{\sin ^{2}(\pi z)}}=\sum _{n=\infty }^{\infty }{\frac {1}{(zn)^{2}}}.}
Using product expansion technique, one can derive

$$
sin
(
π
z
)
=
π
z
∏
n
=
1
∞
(
1
−
z
2
n
2
)
.
{\displaystyle \sin(\pi z)=\pi z\prod _{n=1}^{\infty }\left(1{\frac {z^{2}}{n^{2}}}\right).}
Alternatively, the infinite product for the sine can be proved using complex Fourier series.
Proof of the infinite product for the sine 

Using complex Fourier series, the function
Setting
Therefore, we get
The function
Exponentiating gives
Since
for some open and connected subset of 
Usage of complex sine
[edit]
sin(z) is found in the functional equation for the Gamma function,

$$
Γ
(
s
)
Γ
(
1
−
s
)
=
π
sin
(
π
s
)
,
{\displaystyle \Gamma (s)\Gamma (1s)={\pi \over \sin(\pi s)},}
which in turn is found in the functional equation for the Riemann zetafunction,

$$
ζ
(
s
)
=
2
(
2
π
)
s
−
1
Γ
(
1
−
s
)
sin
(
π
2
s
)
ζ
(
1
−
s
)
.
{\displaystyle \zeta (s)=2(2\pi )^{s1}\Gamma (1s)\sin \left({\frac {\pi }{2}}s\right)\zeta (1s).}
As a holomorphic function, sin
z
is a 2D solution of Laplace’s equation:

$$
Δ
u
(
x
1
,
x
2
)
=
0.
{\displaystyle \Delta u(x_{1},x_{2})=0.}
The complex sine function is also related to the level curves of pendulums.^{[
how?
]}
^{[12]}
^{[
better source needed
]}
Complex graphs
[edit]



real component  imaginary component  magnitude 



real component  imaginary component  magnitude 
History
[edit]
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 function was discovered by Hipparchus of Nicaea (180–125 BCE) and Ptolemy of Roman Egypt (90–165 CE). See in particular Ptolemy’s table of chords.
The function of sine and versine (1 − cosine) can be traced to the
jyā
and
koṭijyā
functions used in Gupta period (320 to 550 CE) Indian astronomy (Aryabhatiya,
Surya Siddhanta), via translation from Sanskrit to Arabic and then from Arabic to Latin.^{[2]}
All six trigonometric functions in current use were known in Islamic mathematics by the 9th century, as was the law of sines, used in solving triangles.^{[13]}
With the exception of the sine (which was adopted from Indian mathematics), the other five modern trigonometric functions were discovered by Arabic mathematicians, including the cosine, tangent, cotangent, secant and cosecant.^{[13]}
AlKhwārizmī (c. 780–850) produced tables of sines, cosines and tangents.^{[14]}
^{[15]}
Muhammad ibn Jābir alHarrānī alBattānī (853–929) discovered the reciprocal functions of secant and cosecant, and produced the first table of cosecants for each degree from 1° to 90°.^{[15]}
The first published use of the abbreviations
sin,
cos, and
tan
is by the 16thcentury 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, Leibniz proved that sin
x
is not an algebraic function of
x.^{[16]}
Roger Cotes computed the derivative of sine in his
Harmonia Mensurarum
(1722).^{[17]}
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 nearmodern abbreviations
sin.,
cos.,
tang.,
cot.,
sec., and
cosec.
^{[18]}
Etymology
[edit]
Etymologically, the word
sine
derives from the Sanskrit word for ‘chord’,
jiva
(
jya
being its more popular synonym). This was transliterated in Arabic as
jiba
(
جيب
), which is however meaningless in that language and abbreviated
jb
(
جب
). Since Arabic is written without short vowels,
jb
was interpreted as the word
jaib
(
جيب
), which means ‘bosom’. When the Arabic texts were translated in the 12th century into medieval Latin by Gerard of Cremona, he used the Latin equivalent for ‘bosom’,
sinus
(which also means ‘bay’ or ‘fold’).^{[19]}
^{[20]}
Gerard was probably not the first scholar to use this translation; Robert of Chester appears to have preceded him and there is evidence of even earlier usage.^{[21]}
The English form
sine
was introduced in the 1590s. The word
cosine
derives from a contraction of the Latin
complementi sinus
.^{[4]}
Software implementations
[edit]
There is no standard algorithm for calculating sine and cosine. IEEE 7542008, the most widely used standard for floatingpoint computation, does not address calculating trigonometric functions such as sine.^{[22]}
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.
sin(10^{22})
.
A common programming optimization, used especially in 3D graphics, is to precalculate a table of sine values, for example one value per degree, then for values inbetween pick the closest precalculated value, or linearly interpolate between the 2 closest values to approximate it. This allows 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 ]}
The CORDIC algorithm is commonly used in scientific calculators.
The sine and cosine functions, along with other trigonometric functions, is widely available across programming languages and platforms. In computing, they are typically abbreviated to
sin
and
cos
.
Some CPU architectures have a builtin instruction for sine, including the Intel x87 FPUs since the 80387.
In programming languages,
sin
and
cos
are typically either a builtin function or found within the language’s standard math library.
For example, the C standard library defines sine functions within math.h:
sin(double)
,
sinf(float)
, and
sinl(long double)
. 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 math.h, such as for cosine, arc sine, and hyperbolic sine (sinh).
Similarly, Python defines
math.sin(x)
and
math.cos(x)
within the builtin
math
module. Complex sine and cosine functions are also available within the
cmath
module, e.g.
cmath.sin(z)
. CPython’s math functions call the C
math
library, and use a doubleprecision floatingpoint format.
Turns based implementations
[edit]
Some software libraries provide implementations of sine and cosine using the input angle in halfturns, a halfturn being an angle of 180 degrees or
radians. Representing angles in turns or halfturns has accuracy advantages and efficiency advantages in some cases.^{[23]}
^{[24]}
In MATLAB, OpenCL, R, Julia, CUDA, and ARM, these function are called
sinpi
and
cospi
.^{[23]}
^{[25]}
^{[24]}
^{[26]}
^{[27]}
^{[28]}
For example,
sinpi(x)
would evaluate to
where
x
is expressed in radians.
The accuracy advantage stems from the ability to perfectly represent key angles like fullturn, halfturn, and quarterturn losslessly in binary floatingpoint or fixedpoint. In contrast, representing
,
, and
in binary floatingpoint or binary scaled fixedpoint always involves a loss of accuracy since irrational numbers cannot be represented with finitely many binary digits.
Turns also have an accuracy advantage and efficiency advantage for computing modulo to one period. Computing modulo 1 turn or modulo 2 halfturns can be losslessly and efficiently computed in both floatingpoint and fixedpoint. For example, computing modulo 1 or modulo 2 for a binary point scaled fixedpoint value requires only a bit shift or bitwise AND operation. In contrast, computing modulo
involves inaccuracies in representing
.
For applications involving angle sensors, the sensor typically provides angle measurements in a form directly compatible with turns or halfturns. For example, an angle sensor may count from 0 to 4096 over one complete revolution.^{[29]}
If halfturns are used as the unit for angle, then the value provided by the sensor directly and losslessly maps to a fixedpoint data type with 11 bits to the right of the binary point. In contrast, if radians are used as the unit for storing the angle, then the inaccuracies and cost of multiplying the raw sensor integer by an approximation to
would be incurred.
See also
[edit]
 Āryabhaṭa’s sine table
 Bhaskara I’s sine approximation formula
 Discrete sine transform
 Euler’s formula
 Generalized trigonometry
 Hyperbolic function
 Dixon elliptic functions
 Lemniscate elliptic functions
 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
 Sinc function
 Sine and cosine transforms
 Sine integral
 Sine quadrant
 Sine wave
 Sine–Gordon equation
 Sinusoidal model
 Trigonometric functions
 Trigonometric integral
Citations
[edit]

^
^{ a }
^{ b }
^{ c }
Weisstein, Eric W. “Sine”.
mathworld.wolfram.com
. Retrieved
20200829
.

^
^{ a }
^{ b }
Uta C. Merzbach, Carl B. Boyer (2011), A History of Mathematics, Hoboken, N.J.: John Wiley & Sons, 3rd ed., p. 189. 
^
Victor J. Katz (2008),
A History of Mathematics, Boston: AddisonWesley, 3rd. ed., p. 253, sidebar 8.1.
“Archived copy”
(PDF). Archived
(PDF)
from the original on 20150414. Retrieved
20150409
.
{{cite web}}
: CS1 maint: archived copy as title (link)

^
^{ a }
^{ b }
“cosine”.

^
^{ a }
^{ b }
“Sine, Cosine, Tangent”.
www.mathsisfun.com
. Retrieved
20200829
.

^
See Ahlfors, pages 43–44. 
^
“Sinesquared function”. Retrieved
August 9,
2019.

^
“OEIS A003957”.
oeis.org
. Retrieved
20190526
.

^
^{ a }
^{ b }
“A105419 – Oeis”.

^
Adlaj, Semjon (2012). “An Eloquent Formula for the Perimeter of an Ellipse”
(PDF).
American Mathematical Society. p. 1097.

^
Rudin, Walter (1987).
Real and Complex Analysis
(Third ed.). McGrawHill Book Company. ISBN0071002766.
p. 299, Theorem 15.4 
^
“Why are the phase portrait of the simple plane pendulum and a domain coloring of sin(z) so similar?”.
math.stackexchange.com
. Retrieved
20190812
.

^
^{ a }
^{ b }
Gingerich, Owen (1986). “Islamic Astronomy”.
Scientific American. Vol. 254. p. 74. Archived from the original on 20131019. Retrieved
20100713
.

^
Jacques Sesiano, “Islamic mathematics”, p. 157, in
Selin, Helaine; D’Ambrosio, Ubiratan, eds. (2000).
Mathematics Across Cultures: The History of Nonwestern Mathematics. Springer Science+Business Media. ISBN9781402002601.

^
^{ a }
^{ b }
“trigonometry”. Encyclopedia Britannica.

^
Nicolás Bourbaki (1994).
Elements of the History of Mathematics
. Springer. ISBN9783540647676.

^
“Why the sine has a simple derivative Archived 20110720 at the Wayback Machine”, in
Historical Notes for Calculus Teachers Archived 20110720 at the Wayback Machine
by V. Frederick Rickey Archived 20110720 at the Wayback Machine 
^
See Merzbach, Boyer (2011). 
^
Eli Maor (1998),
Trigonometric Delights, Princeton: Princeton University Press, p. 3536. 
^
Victor J. Katz (2008),
A History of Mathematics, Boston: AddisonWesley, 3rd. ed., p. 253, sidebar 8.1.
“Archived copy”
(PDF). Archived
(PDF)
from the original on 20150414. Retrieved
20150409
.
{{cite web}}
: CS1 maint: archived copy as title (link)

^
Smith, D.E. (1958) [1925],
History of Mathematics, vol. I, Dover, p. 202, ISBN0486204294

^
Grand Challenges of Informatics, Paul Zimmermann. September 20, 2006 – p. 14/31
“Archived copy”
(PDF). Archived
(PDF)
from the original on 20110716. Retrieved
20100911
.
{{cite web}}
: CS1 maint: archived copy as title (link)

^
^{ a }
^{ b }
“MATLAB Documentation sinpi 
^
^{ a }
^{ b }
“R Documentation sinpi 
^
“OpenCL Documentation sinpi 
^
“Julia Documentation sinpi 
^
“CUDA Documentation sinpi 
^
“ARM Documentation sinpi 
^
“ALLEGRO Angle Sensor Datasheet
References
[edit]

Traupman, Ph.D., John C. (1966),
The New College Latin & English Dictionary
, Toronto: Bantam, ISBN0553276190

Webster’s Seventh New Collegiate Dictionary, Springfield: G. & C. Merriam Company, 1969
External links
[edit]
Look up
sine
in Wiktionary, the free dictionary.

Media related to Sine function at Wikimedia Commons
Cos 2x 3 Sin X 1 0
Sumber: https://en.wikipedia.org/wiki/Sine_and_cosine