Name
Symbol
Formula [nb 1]
Fourier Series
Sine
sin
(
x
)
{\displaystyle \sin(x)}
∑
n
=
0
∞
(
−
1
)
n
x
2
n
+
1
(
2
n
+
1
)
!
{\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n+1}}{(2n+1)!}}}
sin
(
x
)
{\displaystyle \sin(x)}
cas (mathematics)
cas
(
x
)
{\displaystyle \operatorname {cas} (x)}
sin
(
x
)
+
cos
(
x
)
{\displaystyle \sin(x)+\cos(x)}
sin
(
x
)
+
cos
(
x
)
{\displaystyle \sin(x)+\cos(x)}
Cosine
cos
(
x
)
{\displaystyle \cos(x)}
∑
n
=
0
∞
(
−
1
)
n
x
2
n
(
2
n
)
!
{\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n}}{(2n)!}}}
cos
(
x
)
{\displaystyle \cos(x)}
cis (mathematics)
e
i
x
,
cis
(
x
)
{\displaystyle e^{ix},\operatorname {cis} (x)}
cos(x ) + i sin(x )
cos
(
x
)
+
i
sin
(
x
)
{\displaystyle \cos(x)+i\sin(x)}
Tangent
tan
(
x
)
{\displaystyle \tan(x)}
∑
n
=
0
∞
U
2
n
+
1
x
2
n
+
1
(
2
n
+
1
)
!
{\displaystyle \sum _{n=0}^{\infty }{\frac {U_{2n+1}x^{2n+1}}{(2n+1)!}}}
2
∑
n
=
1
∞
(
−
1
)
n
−
1
sin
(
2
n
x
)
{\displaystyle 2\sum _{n=1}^{\infty }(-1)^{n-1}\sin(2nx)}
[1] Cotangent
cot
(
x
)
{\displaystyle \cot(x)}
∑
n
=
0
∞
(
−
1
)
n
2
2
n
B
2
n
x
2
n
−
1
(
2
n
)
!
{\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}2^{2n}B_{2n}x^{2n-1}}{(2n)!}}}
i
+
2
i
∑
n
=
1
∞
(
cos
2
n
x
−
i
sin
2
n
x
)
{\displaystyle i+2i\sum _{n=1}^{\infty }(\cos 2nx-i\sin 2nx)}
[citation needed ] Secant
sec
(
x
)
{\displaystyle \sec(x)}
∑
n
=
0
∞
U
2
n
x
2
n
(
2
n
)
!
{\displaystyle \sum _{n=0}^{\infty }{\frac {U_{2n}x^{2n}}{(2n)!}}}
-
Cosecant
csc
(
x
)
{\displaystyle \csc(x)}
∑
n
=
0
∞
(
−
1
)
n
+
1
2
(
2
2
n
−
1
−
1
)
B
2
n
x
2
n
−
1
(
2
n
)
!
{\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n+1}2\left(2^{2n-1}-1\right)B_{2n}x^{2n-1}}{(2n)!}}}
-
Exsecant
exsec
(
x
)
{\displaystyle \operatorname {exsec} (x)}
sec
(
x
)
−
1
{\displaystyle \sec(x)-1}
-
Excosecant
excsc
(
x
)
{\displaystyle \operatorname {excsc} (x)}
csc
(
x
)
−
1
{\displaystyle \csc(x)-1}
-
Versine
versin
(
x
)
{\displaystyle \operatorname {versin} (x)}
1
−
cos
(
x
)
{\displaystyle 1-\cos(x)}
1
−
cos
(
x
)
{\displaystyle 1-\cos(x)}
Vercosine
vercosin
(
x
)
{\displaystyle \operatorname {vercosin} (x)}
1
+
cos
(
x
)
{\displaystyle 1+\cos(x)}
1
+
cos
(
x
)
{\displaystyle 1+\cos(x)}
Coversine
coversin
(
x
)
{\displaystyle \operatorname {coversin} (x)}
1
−
sin
(
x
)
{\displaystyle 1-\sin(x)}
1
−
sin
(
x
)
{\displaystyle 1-\sin(x)}
Covercosine
covercosin
(
x
)
{\displaystyle \operatorname {covercosin} (x)}
1
+
sin
(
x
)
{\displaystyle 1+\sin(x)}
1
+
sin
(
x
)
{\displaystyle 1+\sin(x)}
Haversine
haversin
(
x
)
{\displaystyle \operatorname {haversin} (x)}
1
−
cos
(
x
)
2
{\displaystyle {\frac {1-\cos(x)}{2}}}
1
2
−
1
2
cos
(
x
)
{\displaystyle {\frac {1}{2}}-{\frac {1}{2}}\cos(x)}
Havercosine
havercosin
(
x
)
{\displaystyle \operatorname {havercosin} (x)}
1
+
cos
(
x
)
2
{\displaystyle {\frac {1+\cos(x)}{2}}}
1
2
+
1
2
cos
(
x
)
{\displaystyle {\frac {1}{2}}+{\frac {1}{2}}\cos(x)}
Hacoversine
hacoversin
(
x
)
{\displaystyle \operatorname {hacoversin} (x)}
1
−
sin
(
x
)
2
{\displaystyle {\frac {1-\sin(x)}{2}}}
1
2
−
1
2
sin
(
x
)
{\displaystyle {\frac {1}{2}}-{\frac {1}{2}}\sin(x)}
Hacovercosine
hacovercosin
(
x
)
{\displaystyle \operatorname {hacovercosin} (x)}
1
+
sin
(
x
)
2
{\displaystyle {\frac {1+\sin(x)}{2}}}
1
2
+
1
2
sin
(
x
)
{\displaystyle {\frac {1}{2}}+{\frac {1}{2}}\sin(x)}
Magnitude of sine wave with amplitude, A, and period, T
-
A
|
sin
(
2
π
T
x
)
|
{\displaystyle A|\sin \left({\frac {2\pi }{T}}x\right)|}
4
A
2
π
+
∑
n
e
v
e
n
−
4
A
π
1
1
−
n
2
cos
(
2
π
n
T
x
)
{\displaystyle {\frac {4A}{2\pi }}+\sum _{n\,\mathrm {even} }{\frac {-4A}{\pi }}{\frac {1}{1-n^{2}}}\cos({\frac {2\pi n}{T}}x)}
[2] : p. 193 Clausen function
Cl
2
(
x
)
{\displaystyle \operatorname {Cl} _{2}(x)}
−
∫
0
x
ln
|
2
sin
x
2
|
d
x
{\displaystyle -\int _{0}^{x}\ln \left|2\sin {\frac {x}{2}}\right|dx}
∑
k
=
1
∞
sin
k
x
k
2
{\displaystyle \sum _{k=1}^{\infty }{\frac {\sin kx}{k^{2}}}}