# Cosines

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##### 3: 4.21 Identities
4.21.8 $\cos u+\cos v=2\cos\left(\frac{u+v}{2}\right)\cos\left(\frac{u-v}{2}\right),$
4.21.16 $2\cos u\cos v=\cos\left(u-v\right)+\cos\left(u+v\right),$
4.21.25 $\cos\left(-z\right)=\cos z,$
4.21.31 $\cos\left(3z\right)=-3\cos z+4{\cos}^{3}z,$
4.21.33 $\cos\left(4z\right)=8{\cos}^{4}z-8{\cos}^{2}z+1.$
##### 4: 4.35 Identities
4.35.7 $\cosh u+\cosh v=2\cosh\left(\frac{u+v}{2}\right)\cosh\left(\frac{u-v}{2}\right),$
4.35.15 $2\cosh u\cosh v=\cosh\left(u+v\right)+\cosh\left(u-v\right),$
4.35.24 $\cosh\left(-z\right)=\cosh z,$
4.35.30 $\cosh\left(3z\right)=-3\cosh z+4{\cosh}^{3}z,$
4.35.35 $\cosh z=\cosh x\cos y+i\sinh x\sin y,$
##### 5: 6.17 Physical Applications
###### §6.17 Physical Applications
Lebedev (1965) gives an application to electromagnetic theory (radiation of a linear half-wave oscillator), in which sine and cosine integrals are used.
##### 6: 4.42 Solution of Triangles
4.42.10 $\sin a\cos B=\cos b\sin c-\sin b\cos c\cos A,$
##### 7: 6.1 Special Notation
Unless otherwise noted, primes indicate derivatives with respect to the argument. The main functions treated in this chapter are the exponential integrals $\operatorname{Ei}\left(x\right)$, $E_{1}\left(z\right)$, and $\operatorname{Ein}\left(z\right)$; the logarithmic integral $\operatorname{li}\left(x\right)$; the sine integrals $\operatorname{Si}\left(z\right)$ and $\operatorname{si}\left(z\right)$; the cosine integrals $\operatorname{Ci}\left(z\right)$ and $\operatorname{Cin}\left(z\right)$.
##### 8: 6.4 Analytic Continuation
###### §6.4 Analytic Continuation
6.4.7 $\mathrm{g}\left(ze^{\pm\pi i}\right)=\mp\pi ie^{\mp iz}+\mathrm{g}\left(z% \right).$
Unless indicated otherwise, in the rest of this chapter and elsewhere in the DLMF the functions $E_{1}\left(z\right)$, $\operatorname{Ci}\left(z\right)$, $\operatorname{Chi}\left(z\right)$, $\mathrm{f}\left(z\right)$, and $\mathrm{g}\left(z\right)$ assume their principal values, that is, the branches that are real on the positive real axis and two-valued on the negative real axis.
##### 9: 4.14 Definitions and Periodicity
4.14.2 $\cos z=\frac{e^{\mathrm{i}z}+e^{-\mathrm{i}z}}{2},$
4.14.6 $\sec z=\frac{1}{\cos z},$
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 $z=\left(k+\tfrac{1}{2}\right)\pi$, $k\in\mathbb{Z}$. …
##### 10: 4.32 Inequalities
4.32.3 $|\cosh x-\cosh y|\geq|x-y|\sqrt{\sinh x\sinh y},$ $x>0$, $y>0$,