# trigonometric

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##### 2: 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).
##### 3: 4.47 Approximations
###### §4.47 Approximations
Clenshaw (1962) and Luke (1975, Chapter 3) give 20D coefficients for $\ln$, $\exp$, $\sin$, $\cos$, $\tan$, $\cot$, $\operatorname{arcsin}$, $\operatorname{arctan}$, $\operatorname{arcsinh}$. Schonfelder (1980) gives 40D coefficients for $\sin$, $\cos$, $\tan$. … Hart et al. (1968) give $\ln$, $\exp$, $\sin$, $\cos$, $\tan$, $\cot$, $\operatorname{arcsin}$, $\operatorname{arccos}$, $\operatorname{arctan}$, $\sinh$, $\cosh$, $\tanh$, $\operatorname{arcsinh}$, $\operatorname{arccosh}$. … Luke (1975, Chapter 3) supplies real and complex approximations for $\ln$, $\exp$, $\sin$, $\cos$, $\tan$, $\operatorname{arctan}$, $\operatorname{arcsinh}$. …
##### 4: 4.1 Special Notation
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. Sometimes in the literature the meanings of $\ln$ and $\operatorname{Ln}$ are interchanged; similarly for $\operatorname{arcsin}z$ and $\operatorname{Arcsin}z$, etc. …${\sin}^{-1}z$ for $\operatorname{arcsin}z$ and $\mathrm{Sin}^{-1}\;z$ for $\operatorname{Arcsin}z$.
##### 5: 4.35 Identities
4.35.14 $2\sinh u\sinh v=\cosh\left(u+v\right)-\cosh\left(u-v\right),$
4.35.16 $2\sinh u\cosh v=\sinh\left(u+v\right)+\sinh\left(u-v\right).$
4.35.18 ${\cosh}^{2}u-{\cosh}^{2}v=\sinh\left(u+v\right)\sinh\left(u-v\right),$
4.35.34 $\sinh z=\sinh x\cos y+i\cosh x\sin y,$
4.35.35 $\cosh z=\cosh x\cos y+i\sinh x\sin y,$
##### 6: 4.14 Definitions and Periodicity
###### §4.14 Definitions and Periodicity
4.14.7 $\cot z=\frac{\cos z}{\sin z}=\frac{1}{\tan z}.$
The functions $\sin z$ and $\cos z$ are entire. In $\mathbb{C}$ the zeros of $\sin z$ are $z=k\pi$, $k\in\mathbb{Z}$; the zeros of $\cos z$ are $z=\left(k+\tfrac{1}{2}\right)\pi$, $k\in\mathbb{Z}$. The functions $\tan z$, $\csc z$, $\sec z$, and $\cot z$ are meromorphic, and the locations of their zeros and poles follow from (4.14.4) to (4.14.7). …
##### 7: 4.21 Identities
###### §4.21 Identities
The functions $\sinh z$ and $\cosh z$ have period $2\pi i$, and $\tanh z$ has period $\pi i$. The zeros of $\sinh z$ and $\cosh z$ are $z=ik\pi$ and $z=i\left(k+\frac{1}{2}\right)\pi$, respectively, $k\in\mathbb{Z}$.
4.32.3 $|\cosh x-\cosh y|\geq|x-y|\sqrt{\sinh x\sinh y},$ $x>0$, $y>0$,
4.32.4 $\operatorname{arctan}x\leq\tfrac{1}{2}\pi\tanh x,$ $x\geq 0$.