# trigonometric arguments

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## 1—10 of 65 matching pages

##### 1: 4.15 Graphics
###### §4.15(iii) Complex Arguments: Surfaces
The corresponding surfaces for $\operatorname{arccos}\left(x+iy\right)$, $\operatorname{arccot}\left(x+iy\right)$, $\operatorname{arcsec}\left(x+iy\right)$ can be visualized from Figures 4.15.9, 4.15.11, 4.15.13 with the aid of equations (4.23.16)–(4.23.18).
##### 3: 4.45 Methods of Computation
The inverses $\operatorname{arcsinh}$, $\operatorname{arccosh}$, and $\operatorname{arctanh}$ can be computed from the logarithmic forms given in §4.37(iv), with real arguments. …
##### 4: 23.12 Asymptotic Approximations
23.12.2 $\zeta\left(z\right)=\frac{\pi^{2}}{4\omega_{1}^{2}}\left(\frac{z}{3}+\frac{2% \omega_{1}}{\pi}\cot\left(\frac{\pi z}{2\omega_{1}}\right)-8\left(z-\frac{% \omega_{1}}{\pi}\sin\left(\frac{\pi z}{\omega_{1}}\right)\right)q^{2}+O\left(q% ^{4}\right)\right),$
##### 6: 4.29 Graphics
###### §4.29(i) Real Arguments Figure 4.29.6: Principal values of arccsch ⁡ x and arcsech ⁡ x . … Magnify
###### §4.29(ii) Complex Arguments
The conformal mapping $w=\sinh z$ is obtainable from Figure 4.15.7 by rotating both the $w$-plane and the $z$-plane through an angle $\frac{1}{2}\pi$, compare (4.28.8). The surfaces for the complex hyperbolic and inverse hyperbolic functions are similar to the surfaces depicted in §4.15(iii) for the trigonometric and inverse trigonometric functions. …
##### 7: 19.7 Connection Formulas
$\sin\theta=\frac{\sqrt{1+k^{2}}\sin\phi}{\sqrt{1+k^{2}{\sin}^{2}\phi}},$
###### Imaginary-Argument Transformation
With $\sinh\phi=\tan\psi$, … The first of the three relations maps each circular region onto itself and each hyperbolic region onto the other; in particular, it gives the Cauchy principal value of $\Pi\left(\phi,\alpha^{2},k\right)$ when $\alpha^{2}>{\csc}^{2}\phi$ (see (19.6.5) for the complete case). Let $c={\csc}^{2}\phi\neq\alpha^{2}$. …
19.11.2 $E\left(\theta,k\right)+E\left(\phi,k\right)=E\left(\psi,k\right)+k^{2}\sin% \theta\sin\phi\sin\psi.$
19.11.8 $E\left(\phi,k\right)=E\left(k\right)-E\left(\theta,k\right)+k^{2}\sin\theta% \sin\phi,$
##### 9: 19.14 Reduction of General Elliptic Integrals
19.14.5 ${\sin}^{2}\phi=\frac{\gamma-\alpha}{U^{2}+\gamma},$
19.14.7 ${\sin}^{2}\phi=\frac{(\gamma-\alpha)x^{2}}{a_{1}a_{2}+\gamma x^{2}}.$
19.14.8 ${\sin}^{2}\phi=\frac{\gamma-\alpha}{b_{1}b_{2}y^{2}+\gamma}.$
19.14.9 ${\sin}^{2}\phi=\frac{(\gamma-\alpha)(x^{2}-y^{2})}{\gamma(x^{2}-y^{2})-a_{1}(a% _{2}+b_{2}x^{2})}.$
19.14.10 ${\sin}^{2}\phi=\frac{(\gamma-\alpha)(y^{2}-x^{2})}{\gamma(y^{2}-x^{2})-a_{1}(a% _{2}+b_{2}y^{2})}.$
##### 10: 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(iii) Multiples of the Argument
4.35.34 $\sinh z=\sinh x\cos y+i\cosh x\sin y,$
4.35.35 $\cosh z=\cosh x\cos y+i\sinh x\sin y,$