# trigonometric functions

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##### 1: 4.14 Definitions and Periodicity
4.14.2 $\cos z=\frac{e^{\mathrm{i}z}+e^{-\mathrm{i}z}}{2},$
4.14.4 $\tan z=\frac{\sin z}{\cos z},$
4.14.5 $\csc z=\frac{1}{\sin z},$
4.14.6 $\sec z=\frac{1}{\cos z},$
4.14.7 $\cot z=\frac{\cos z}{\sin z}=\frac{1}{\tan z}.$
##### 2: 4.28 Definitions and Periodicity
4.28.2 $\cosh z=\frac{e^{z}+e^{-z}}{2},$
4.28.4 $\tanh z=\frac{\sinh z}{\cosh z},$
4.28.5 $\operatorname{csch}z=\frac{1}{\sinh z},$
4.28.6 $\operatorname{sech}z=\frac{1}{\cosh z},$
4.28.7 $\coth z=\frac{1}{\tanh z}.$
##### 3: 4.23 Inverse Trigonometric Functions
###### §4.23 Inverse TrigonometricFunctions
4.23.7 $\operatorname{arccsc}z=\operatorname{arcsin}\left(1/z\right),$
4.23.8 $\operatorname{arcsec}z=\operatorname{arccos}\left(1/z\right).$
4.23.9 $\operatorname{arccot}z=\operatorname{arctan}\left(1/z\right),$ $z\neq\pm\mathrm{i}$.
##### 4: 4.37 Inverse Hyperbolic Functions
4.37.4 $\operatorname{Arccsch}z=\operatorname{Arcsinh}\left(1/z\right),$
4.37.7 $\operatorname{arccsch}z=\operatorname{arcsinh}\left(1/z\right),$
4.37.8 $\operatorname{arcsech}z=\operatorname{arccosh}\left(1/z\right).$
4.37.9 $\operatorname{arccoth}z=\operatorname{arctanh}\left(1/z\right),$ $z\neq\pm 1$.
##### 5: 17.3 $q$-Elementary and $q$-Special Functions
17.3.3 $\mathrm{sin}_{q}\left(x\right)=\frac{1}{2i}(e_{q}\left(ix\right)-e_{q}\left(-% ix\right))=\sum_{n=0}^{\infty}\frac{(1-q)^{2n+1}(-1)^{n}x^{2n+1}}{\left(q;q% \right)_{2n+1}},$
17.3.4 $\mathrm{Sin}_{q}\left(x\right)=\frac{1}{2i}(E_{q}\left(ix\right)-E_{q}\left(-% ix\right))=\sum_{n=0}^{\infty}\frac{(1-q)^{2n+1}q^{n(2n+1)}(-1)^{n}x^{2n+1}}{% \left(q;q\right)_{2n+1}}.$
17.3.5 $\mathrm{cos}_{q}\left(x\right)=\frac{1}{2}(e_{q}\left(ix\right)+e_{q}\left(-ix% \right))=\sum_{n=0}^{\infty}\frac{(1-q)^{2n}(-1)^{n}x^{2n}}{\left(q;q\right)_{% 2n}},$
17.3.6 $\mathrm{Cos}_{q}\left(x\right)=\frac{1}{2}(E_{q}\left(ix\right)+E_{q}\left(-ix% \right))=\sum_{n=0}^{\infty}\frac{(1-q)^{2n}q^{n(2n-1)}(-1)^{n}x^{2n}}{\left(q% ;q\right)_{2n}}.$
##### 6: 6.2 Definitions and Interrelations
This is also true of the functions $\mathrm{Ci}\left(z\right)$ and $\mathrm{Chi}\left(z\right)$ defined in §6.2(ii). … $\mathrm{Si}\left(z\right)$ is an odd entire function. …$\mathrm{Cin}\left(z\right)$ is an even entire function. …
6.2.17 $\mathrm{f}\left(z\right)=\phantom{+}\mathrm{Ci}\left(z\right)\sin z-\mathrm{si% }\left(z\right)\cos z,$
6.2.18 $\mathrm{g}\left(z\right)=-\mathrm{Ci}\left(z\right)\cos z-\mathrm{si}\left(z% \right)\sin z.$
##### 7: 7.2 Definitions
7.2.7 $C\left(z\right)=\int_{0}^{z}\cos\left(\tfrac{1}{2}\pi t^{2}\right)\mathrm{d}t,$
7.2.8 $S\left(z\right)=\int_{0}^{z}\sin\left(\tfrac{1}{2}\pi t^{2}\right)\mathrm{d}t,$
$\mathcal{F}\left(z\right)$, $C\left(z\right)$, and $S\left(z\right)$ are entire functions of $z$, as are $\mathrm{f}\left(z\right)$ and $\mathrm{g}\left(z\right)$ in the next subsection. …
7.2.10 $\mathrm{f}\left(z\right)=\left(\tfrac{1}{2}-S\left(z\right)\right)\cos\left(% \tfrac{1}{2}\pi z^{2}\right)-\left(\tfrac{1}{2}-C\left(z\right)\right)\sin% \left(\tfrac{1}{2}\pi z^{2}\right),$
7.2.11 $\mathrm{g}\left(z\right)=\left(\tfrac{1}{2}-C\left(z\right)\right)\cos\left(% \tfrac{1}{2}\pi z^{2}\right)+\left(\tfrac{1}{2}-S\left(z\right)\right)\sin% \left(\tfrac{1}{2}\pi z^{2}\right).$
##### 8: 4.27 Sums
###### §4.27 Sums
For sums of trigonometric and inverse trigonometric functions see Gradshteyn and Ryzhik (2000, Chapter 1), Hansen (1975, §§14–42), Oberhettinger (1973), and Prudnikov et al. (1986a, Chapter 5).
##### 9: 4.1 Special Notation
 $k,m,n$ integers. …
The main purpose of the present chapter is to extend these definitions and properties to complex arguments $z$. The main functions treated in this chapter are the logarithm $\ln z$, $\operatorname{Ln}z$; the exponential $\exp z$, $e^{z}$; the circular trigonometric (or just trigonometric) functions $\sin z$, $\cos z$, $\tan z$, $\csc z$, $\sec z$, $\cot z$; the inverse trigonometric functions $\operatorname{arcsin}z$, $\operatorname{Arcsin}z$, etc. ; the hyperbolic trigonometric (or just hyperbolic) functions $\sinh z$, $\cosh z$, $\tanh z$, $\operatorname{csch}z$, $\operatorname{sech}z$, $\coth z$; the inverse hyperbolic functions $\operatorname{arcsinh}z$, $\operatorname{Arcsinh}z$, etc. …