# trigonometric functions

(0.016 seconds)

## 11—20 of 332 matching pages

##### 11: 4.15 Graphics
###### §4.15(i) Real Arguments Figure 4.15.7: Conformal mapping of sine and inverse sine. … Magnify
###### §4.15(iii) Complex Arguments: Surfaces
4.15.1 $\cos\left(x+iy\right)=\sin\left(x+\tfrac{1}{2}\pi+iy\right),$
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).
##### 12: 22.10 Maclaurin Series
22.10.7 $\operatorname{sn}\left(z,k\right)=\tanh z-\frac{{k^{\prime}}^{2}}{4}(z-\sinh z% \cosh z){\operatorname{sech}^{2}}z+O\left({k^{\prime}}^{4}\right),$
22.10.8 $\operatorname{cn}\left(z,k\right)=\operatorname{sech}z+\frac{{k^{\prime}}^{2}}% {4}(z-\sinh z\cosh z)\tanh z\operatorname{sech}z+O\left({k^{\prime}}^{4}\right),$
22.10.9 $\operatorname{dn}\left(z,k\right)=\operatorname{sech}z+\frac{{k^{\prime}}^{2}}% {4}(z+\sinh z\cosh z)\tanh z\operatorname{sech}z+O\left({k^{\prime}}^{4}\right).$
##### 13: 4.35 Identities
4.35.34 $\sinh z=\sinh x\cos y+i\cosh x\sin y,$
4.35.35 $\cosh z=\cosh x\cos y+i\sinh x\sin y,$
4.35.36 $\tanh z=\frac{\sinh\left(2x\right)+i\sin\left(2y\right)}{\cosh\left(2x\right)+% \cos\left(2y\right)},$
4.35.37 $\coth z=\frac{\sinh\left(2x\right)-i\sin\left(2y\right)}{\cosh\left(2x\right)-% \cos\left(2y\right)}.$
4.35.38 $|\sinh z|=({\sinh^{2}}x+{\sin^{2}}y)^{1/2}=\left(\tfrac{1}{2}(\cosh\left(2x% \right)-\cos\left(2y\right))\right)^{1/2},$
##### 14: 4.42 Solution of Triangles
###### §4.42 Solution of Triangles
4.42.11 $\cos a\cos C=\sin a\cot b-\sin C\cot B,$
##### 15: 4.20 Derivatives and Differential Equations
###### §4.20 Derivatives and Differential Equations
With $a\neq 0$, the general solutions of the differential equations …
4.20.12 $w=A\cos\left(az\right)+B\sin\left(az\right),$
##### 16: 4.17 Special Values and Limits
4.17.1 $\lim_{z\to 0}\frac{\sin z}{z}=1,$
4.17.2 $\lim_{z\to 0}\frac{\tan z}{z}=1.$
##### 17: 4.31 Special Values and Limits
4.31.1 $\lim_{z\to 0}\frac{\sinh z}{z}=1,$
4.31.2 $\lim_{z\to 0}\frac{\tanh z}{z}=1,$
4.31.3 $\lim_{z\to 0}\frac{\cosh z-1}{z^{2}}=\frac{1}{2}.$
##### 19: 4.29 Graphics
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. …
##### 20: 4.18 Inequalities
###### §4.18 Inequalities
4.18.3 $\cos x\leq\frac{\sin x}{x}\leq 1,$ $0\leq x\leq\pi$,