4 Elementary Functions Trigonometric Functions 4.20 Derivatives and Differential Equations 4.22 Infinite Products and Partial Fractions

§4.21 Identities

ⓘ

Keywords:: identities, trigonometric functions
Permalink:: http://dlmf.nist.gov/4.21
See also:: Annotations for Ch.4

§4.21(i) Addition Formulas

ⓘ

Keywords:: addition formulas, trigonometric functions
Notes:: See Hobson (1928, Chapter 4). For (4.21.1) use (4.21.2) and (4.21.3) with $\nu=\tfrac{1}{4}\pi$ .
Permalink:: http://dlmf.nist.gov/4.21.i
Addition (effective with 1.1.0):: Equation (4.21.1_5) was added.
Suggested 2018-08-14 by Ted Ersek
See also:: Annotations for §4.21 and Ch.4

4.21.1		$\sin u\pm\cos u=\sqrt{2}\sin\left(u\pm\tfrac{1}{4}\pi\right)=\pm\sqrt{2}\cos% \left(u\mp\tfrac{1}{4}\pi\right).$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function and $\sin\NVar{z}$ : sine function Referenced by: §4.21(i), Erratum (V1.0.7) for Equation (4.21.1) Permalink: http://dlmf.nist.gov/4.21.E1 Encodings: TeX, pMML, png Correction (effective with 1.0.7): Originally the symbol $\pm$ was missing after the second equal sign. Suggested 2012-09-27 by Dennis M. Heim See also: Annotations for §4.21(i), §4.21 and Ch.4

4.21.1_5		$A\cos u+B\sin u=\sqrt{A^{2}+B^{2}}\cos\left(u-\operatorname{ph}\left(A+B% \mathrm{i}\right)\right),$
$A,B\in\mathbb{R}$ ,
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\in$ : element of, $\mathrm{i}$ : imaginary unit, $\operatorname{ph}$ : phase, $\mathbb{R}$ : real line and $\sin\NVar{z}$ : sine function Referenced by: §4.21(i), Erratum (V1.1.0) for Additions Permalink: http://dlmf.nist.gov/4.21.E1_5 Encodings: TeX, pMML, png Addition (effective with 1.1.0): This equation was added. Suggested 2018-08-14 by Ted Ersek See also: Annotations for §4.21(i), §4.21 and Ch.4

4.21.2	$\displaystyle\sin\left(u\pm v\right)$	$\displaystyle=\sin u\cos v\pm\cos u\sin v,$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function and $\sin\NVar{z}$ : sine function A&S Ref: 4.3.16 Referenced by: §4.21(i), (9.8.13) Permalink: http://dlmf.nist.gov/4.21.E2 Encodings: TeX, pMML, png See also: Annotations for §4.21(i), §4.21 and Ch.4
4.21.3	$\displaystyle\cos\left(u\pm v\right)$	$\displaystyle=\cos u\cos v\mp\sin u\sin v,$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function and $\sin\NVar{z}$ : sine function A&S Ref: 4.3.17 Referenced by: §4.21(i) Permalink: http://dlmf.nist.gov/4.21.E3 Encodings: TeX, pMML, png See also: Annotations for §4.21(i), §4.21 and Ch.4
4.21.4	$\displaystyle\tan\left(u\pm v\right)$	$\displaystyle=\frac{\tan u\pm\tan v}{1\mp\tan u\tan v},$
ⓘ Symbols: $\tan\NVar{z}$ : tangent function A&S Ref: 4.3.18 Referenced by: (9.8.17) Permalink: http://dlmf.nist.gov/4.21.E4 Encodings: TeX, pMML, png See also: Annotations for §4.21(i), §4.21 and Ch.4
4.21.5	$\displaystyle\cot\left(u\pm v\right)$	$\displaystyle=\frac{\pm\cot u\cot v-1}{\cot u\pm\cot v}.$
ⓘ Symbols: $\cot\NVar{z}$ : cotangent function A&S Ref: 4.3.19 Permalink: http://dlmf.nist.gov/4.21.E5 Encodings: TeX, pMML, png See also: Annotations for §4.21(i), §4.21 and Ch.4

4.21.6	$\displaystyle\sin u+\sin v$	$\displaystyle=2\sin\left(\frac{u+v}{2}\right)\cos\left(\frac{u-v}{2}\right),$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function and $\sin\NVar{z}$ : sine function A&S Ref: 4.3.34 Permalink: http://dlmf.nist.gov/4.21.E6 Encodings: TeX, pMML, png See also: Annotations for §4.21(i), §4.21 and Ch.4
4.21.7	$\displaystyle\sin u-\sin v$	$\displaystyle=2\cos\left(\frac{u+v}{2}\right)\sin\left(\frac{u-v}{2}\right),$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function and $\sin\NVar{z}$ : sine function A&S Ref: 4.3.35 Permalink: http://dlmf.nist.gov/4.21.E7 Encodings: TeX, pMML, png See also: Annotations for §4.21(i), §4.21 and Ch.4
4.21.8	$\displaystyle\cos u+\cos v$	$\displaystyle=2\cos\left(\frac{u+v}{2}\right)\cos\left(\frac{u-v}{2}\right),$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function A&S Ref: 4.3.36 Permalink: http://dlmf.nist.gov/4.21.E8 Encodings: TeX, pMML, png See also: Annotations for §4.21(i), §4.21 and Ch.4
4.21.9	$\displaystyle\cos u-\cos v$	$\displaystyle=-2\sin\left(\frac{u+v}{2}\right)\sin\left(\frac{u-v}{2}\right).$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function and $\sin\NVar{z}$ : sine function A&S Ref: 4.3.37 Permalink: http://dlmf.nist.gov/4.21.E9 Encodings: TeX, pMML, png See also: Annotations for §4.21(i), §4.21 and Ch.4

4.21.10	$\displaystyle\tan u\pm\tan v$	$\displaystyle=\frac{\sin\left(u\pm v\right)}{\cos u\cos v},$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function and $\tan\NVar{z}$ : tangent function A&S Ref: 4.3.38 Permalink: http://dlmf.nist.gov/4.21.E10 Encodings: TeX, pMML, png See also: Annotations for §4.21(i), §4.21 and Ch.4
4.21.11	$\displaystyle\cot u\pm\cot v$	$\displaystyle=\frac{\sin\left(v\pm u\right)}{\sin u\sin v}.$
ⓘ Symbols: $\cot\NVar{z}$ : cotangent function and $\sin\NVar{z}$ : sine function A&S Ref: 4.3.39 Permalink: http://dlmf.nist.gov/4.21.E11 Encodings: TeX, pMML, png See also: Annotations for §4.21(i), §4.21 and Ch.4

§4.21(ii) Squares and Products

ⓘ

Keywords:: squares and products, trigonometric functions
Notes:: See Hobson (1928, pp. 19, 45, 60).
Permalink:: http://dlmf.nist.gov/4.21.ii
See also:: Annotations for §4.21 and Ch.4

4.21.12		${\sin}^{2}z+{\cos}^{2}z=1,$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function and $z$ : complex variable A&S Ref: 4.3.10 Permalink: http://dlmf.nist.gov/4.21.E12 Encodings: TeX, pMML, png See also: Annotations for §4.21(ii), §4.21 and Ch.4

4.21.13		${\sec}^{2}z=1+{\tan}^{2}z,$
ⓘ Symbols: $\sec\NVar{z}$ : secant function, $\tan\NVar{z}$ : tangent function and $z$ : complex variable A&S Ref: 4.3.11 Permalink: http://dlmf.nist.gov/4.21.E13 Encodings: TeX, pMML, png See also: Annotations for §4.21(ii), §4.21 and Ch.4

4.21.14		${\csc}^{2}z=1+{\cot}^{2}z.$
ⓘ Symbols: $\csc\NVar{z}$ : cosecant function, $\cot\NVar{z}$ : cotangent function and $z$ : complex variable A&S Ref: 4.3.12 Permalink: http://dlmf.nist.gov/4.21.E14 Encodings: TeX, pMML, png See also: Annotations for §4.21(ii), §4.21 and Ch.4

4.21.15		$2\sin u\sin v=\cos\left(u-v\right)-\cos\left(u+v\right),$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function and $\sin\NVar{z}$ : sine function A&S Ref: 4.3.31 Permalink: http://dlmf.nist.gov/4.21.E15 Encodings: TeX, pMML, png See also: Annotations for §4.21(ii), §4.21 and Ch.4

4.21.16		$2\cos u\cos v=\cos\left(u-v\right)+\cos\left(u+v\right),$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function A&S Ref: 4.3.32 Permalink: http://dlmf.nist.gov/4.21.E16 Encodings: TeX, pMML, png See also: Annotations for §4.21(ii), §4.21 and Ch.4

4.21.17		$2\sin u\cos v=\sin\left(u-v\right)+\sin\left(u+v\right).$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function and $\sin\NVar{z}$ : sine function A&S Ref: 4.3.33 Permalink: http://dlmf.nist.gov/4.21.E17 Encodings: TeX, pMML, png See also: Annotations for §4.21(ii), §4.21 and Ch.4

4.21.18	$\displaystyle{\sin}^{2}u-{\sin}^{2}v$	$\displaystyle=\sin\left(u+v\right)\sin\left(u-v\right),$
ⓘ Symbols: $\sin\NVar{z}$ : sine function A&S Ref: 4.3.40 Permalink: http://dlmf.nist.gov/4.21.E18 Encodings: TeX, pMML, png See also: Annotations for §4.21(ii), §4.21 and Ch.4
4.21.19	$\displaystyle{\cos}^{2}u-{\cos}^{2}v$	$\displaystyle=-\sin\left(u+v\right)\sin\left(u-v\right),$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function and $\sin\NVar{z}$ : sine function A&S Ref: 4.3.41 Permalink: http://dlmf.nist.gov/4.21.E19 Encodings: TeX, pMML, png See also: Annotations for §4.21(ii), §4.21 and Ch.4
4.21.20	$\displaystyle{\cos}^{2}u-{\sin}^{2}v$	$\displaystyle=\cos\left(u+v\right)\cos\left(u-v\right).$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function and $\sin\NVar{z}$ : sine function A&S Ref: 4.3.42 Permalink: http://dlmf.nist.gov/4.21.E20 Encodings: TeX, pMML, png See also: Annotations for §4.21(ii), §4.21 and Ch.4

§4.21(iii) Multiples of the Argument

ⓘ

Keywords:: multiples of argument, trigonometric functions
Notes:: See Hobson (1928, pp. 21, 52–53, 63–69, 237–239). For (4.21.35) see Walker (1996, §1.9).
Permalink:: http://dlmf.nist.gov/4.21.iii
See also:: Annotations for §4.21 and Ch.4

4.21.21		$\sin\frac{z}{2}=\pm\left(\frac{1-\cos z}{2}\right)^{1/2},$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function and $z$ : complex variable A&S Ref: 4.3.20 Referenced by: §4.21(iii) Permalink: http://dlmf.nist.gov/4.21.E21 Encodings: TeX, pMML, png See also: Annotations for §4.21(iii), §4.21 and Ch.4

4.21.22		$\cos\frac{z}{2}=\pm\left(\frac{1+\cos z}{2}\right)^{1/2},$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function and $z$ : complex variable A&S Ref: 4.3.21 Permalink: http://dlmf.nist.gov/4.21.E22 Encodings: TeX, pMML, png See also: Annotations for §4.21(iii), §4.21 and Ch.4

4.21.23		$\tan\frac{z}{2}=\pm\left(\frac{1-\cos z}{1+\cos z}\right)^{1/2}=\frac{1-\cos z% }{\sin z}=\frac{\sin z}{1+\cos z}.$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $\tan\NVar{z}$ : tangent function and $z$ : complex variable A&S Ref: 4.3.22 Referenced by: §4.21(iii) Permalink: http://dlmf.nist.gov/4.21.E23 Encodings: TeX, pMML, png See also: Annotations for §4.21(iii), §4.21 and Ch.4

In (4.21.21)–(4.21.23) Table 4.16.1 and analytic continuation will assist in resolving sign ambiguities.

4.21.24	$\displaystyle\sin\left(-z\right)$	$\displaystyle=-\sin z,$
ⓘ Symbols: $\sin\NVar{z}$ : sine function and $z$ : complex variable A&S Ref: 4.3.13 Permalink: http://dlmf.nist.gov/4.21.E24 Encodings: TeX, pMML, png See also: Annotations for §4.21(iii), §4.21 and Ch.4
4.21.25	$\displaystyle\cos\left(-z\right)$	$\displaystyle=\cos z,$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function and $z$ : complex variable A&S Ref: 4.3.14 Permalink: http://dlmf.nist.gov/4.21.E25 Encodings: TeX, pMML, png See also: Annotations for §4.21(iii), §4.21 and Ch.4
4.21.26	$\displaystyle\tan\left(-z\right)$	$\displaystyle=-\tan z.$
ⓘ Symbols: $\tan\NVar{z}$ : tangent function and $z$ : complex variable A&S Ref: 4.3.15 Permalink: http://dlmf.nist.gov/4.21.E26 Encodings: TeX, pMML, png See also: Annotations for §4.21(iii), §4.21 and Ch.4

4.21.27		$\sin\left(2z\right)=2\sin z\cos z=\frac{2\tan z}{1+{\tan}^{2}z},$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $\tan\NVar{z}$ : tangent function and $z$ : complex variable A&S Ref: 4.3.24 Permalink: http://dlmf.nist.gov/4.21.E27 Encodings: TeX, pMML, png See also: Annotations for §4.21(iii), §4.21 and Ch.4

4.21.28		$\cos\left(2z\right)=2{\cos}^{2}z-1=1-2{\sin}^{2}z={\cos}^{2}z-{\sin}^{2}z=% \frac{1-{\tan}^{2}z}{1+{\tan}^{2}z},$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $\tan\NVar{z}$ : tangent function and $z$ : complex variable A&S Ref: 4.3.25 Permalink: http://dlmf.nist.gov/4.21.E28 Encodings: TeX, pMML, png See also: Annotations for §4.21(iii), §4.21 and Ch.4

4.21.29		$\tan\left(2z\right)=\frac{2\tan z}{1-{\tan}^{2}z}=\frac{2\cot z}{{\cot}^{2}z-1% }=\frac{2}{\cot z-\tan z}.$
ⓘ Symbols: $\cot\NVar{z}$ : cotangent function, $\tan\NVar{z}$ : tangent function and $z$ : complex variable A&S Ref: 4.3.26 Permalink: http://dlmf.nist.gov/4.21.E29 Encodings: TeX, pMML, png See also: Annotations for §4.21(iii), §4.21 and Ch.4

4.21.30	$\displaystyle\sin\left(3z\right)$	$\displaystyle=3\sin z-4{\sin}^{3}z,$
ⓘ Symbols: $\sin\NVar{z}$ : sine function and $z$ : complex variable A&S Ref: 4.3.27 Permalink: http://dlmf.nist.gov/4.21.E30 Encodings: TeX, pMML, png See also: Annotations for §4.21(iii), §4.21 and Ch.4
4.21.31	$\displaystyle\cos\left(3z\right)$	$\displaystyle=-3\cos z+4{\cos}^{3}z,$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function and $z$ : complex variable A&S Ref: 4.3.28 Permalink: http://dlmf.nist.gov/4.21.E31 Encodings: TeX, pMML, png See also: Annotations for §4.21(iii), §4.21 and Ch.4
4.21.32	$\displaystyle\sin\left(4z\right)$	$\displaystyle=8{\cos}^{3}z\sin z-4\cos z\sin z,$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function and $z$ : complex variable A&S Ref: 4.3.29 Permalink: http://dlmf.nist.gov/4.21.E32 Encodings: TeX, pMML, png See also: Annotations for §4.21(iii), §4.21 and Ch.4
4.21.33	$\displaystyle\cos\left(4z\right)$	$\displaystyle=8{\cos}^{4}z-8{\cos}^{2}z+1.$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function and $z$ : complex variable A&S Ref: 4.3.30 Permalink: http://dlmf.nist.gov/4.21.E33 Encodings: TeX, pMML, png See also: Annotations for §4.21(iii), §4.21 and Ch.4

De Moivre’s Theorem

ⓘ

Keywords:: De Moivre’s theorem, trigonometric functions
See also:: Annotations for §4.21(iii), §4.21 and Ch.4

When $n\in\mathbb{Z}$

4.21.34		$\cos\left(nz\right)+i\sin\left(nz\right)=(\cos z+i\sin z)^{n}.$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\mathrm{i}$ : imaginary unit, $\sin\NVar{z}$ : sine function, $n$ : integer and $z$ : complex variable A&S Ref: 4.3.48 Permalink: http://dlmf.nist.gov/4.21.E34 Encodings: TeX, pMML, png See also: Annotations for §4.21(iii), §4.21(iii), §4.21 and Ch.4

This result is also valid when $n$ is fractional or complex, provided that $-\pi\leq\Re z\leq\pi$ .

4.21.35		$\sin\left(nz\right)=2^{n-1}\prod_{k=0}^{n-1}\sin\left(z+\frac{k\pi}{n}\right),$
$n=1,2,3,\dots$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\sin\NVar{z}$ : sine function, $k$ : integer, $n$ : integer and $z$ : complex variable Referenced by: §23.10(iii), §4.21(iii) Permalink: http://dlmf.nist.gov/4.21.E35 Encodings: TeX, pMML, png See also: Annotations for §4.21(iii), §4.21(iii), §4.21 and Ch.4

If $t=\tan\left(\frac{1}{2}z\right)$ , then

4.21.36	$\displaystyle\sin z$	$\displaystyle=\frac{2t}{1+t^{2}},$
	$\displaystyle\cos z$	$\displaystyle=\frac{1-t^{2}}{1+t^{2}},$
	$\displaystyle\,\mathrm{d}z$	$\displaystyle=\frac{2}{1+t^{2}}\,\mathrm{d}t.$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\sin\NVar{z}$ : sine function and $z$ : complex variable A&S Ref: 4.3.23 Permalink: http://dlmf.nist.gov/4.21.E36 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §4.21(iii), §4.21(iii), §4.21 and Ch.4

§4.21(iv) Real and Imaginary Parts; Moduli

ⓘ

Keywords:: moduli, real and imaginary parts, trigonometric functions
Notes:: See Hobson (1928, p. 331).
Permalink:: http://dlmf.nist.gov/4.21.iv
See also:: Annotations for §4.21 and Ch.4

With $z=x+iy$

4.21.37		$\sin z=\sin x\cosh y+\mathrm{i}\cos x\sinh y,$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\sin\NVar{z}$ : sine function, $x$ : real variable, $y$ : real variable and $z$ : complex variable A&S Ref: 4.3.55 Permalink: http://dlmf.nist.gov/4.21.E37 Encodings: TeX, pMML, png See also: Annotations for §4.21(iv), §4.21 and Ch.4

4.21.38		$\cos z=\cos x\cosh y-\mathrm{i}\sin x\sinh y,$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\sin\NVar{z}$ : sine function, $x$ : real variable, $y$ : real variable and $z$ : complex variable A&S Ref: 4.3.56 Permalink: http://dlmf.nist.gov/4.21.E38 Encodings: TeX, pMML, png See also: Annotations for §4.21(iv), §4.21 and Ch.4

4.21.39		$\tan z=\frac{\sin\left(2x\right)+\mathrm{i}\sinh\left(2y\right)}{\cos\left(2x% \right)+\cosh\left(2y\right)},$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\sin\NVar{z}$ : sine function, $\tan\NVar{z}$ : tangent function, $x$ : real variable, $y$ : real variable and $z$ : complex variable A&S Ref: 4.3.57 Permalink: http://dlmf.nist.gov/4.21.E39 Encodings: TeX, pMML, png See also: Annotations for §4.21(iv), §4.21 and Ch.4

4.21.40		$\cot z=\frac{\sin\left(2x\right)-\mathrm{i}\sinh\left(2y\right)}{\cosh\left(2y% \right)-\cos\left(2x\right)}.$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\cot\NVar{z}$ : cotangent function, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\sin\NVar{z}$ : sine function, $x$ : real variable, $y$ : real variable and $z$ : complex variable A&S Ref: 4.3.58 Permalink: http://dlmf.nist.gov/4.21.E40 Encodings: TeX, pMML, png See also: Annotations for §4.21(iv), §4.21 and Ch.4

4.21.41		$\|\sin z\|=({\sin}^{2}x+{\sinh}^{2}y)^{1/2}=\left(\tfrac{1}{2}\left(\cosh\left(2% y\right)-\cos\left(2x\right)\right)\right)^{1/2},$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\sin\NVar{z}$ : sine function, $x$ : real variable, $y$ : real variable and $z$ : complex variable A&S Ref: 4.3.59 Permalink: http://dlmf.nist.gov/4.21.E41 Encodings: TeX, pMML, png See also: Annotations for §4.21(iv), §4.21 and Ch.4

4.21.42		$\|\cos z\|=({\cos}^{2}x+{\sinh}^{2}y)^{1/2}=\left(\tfrac{1}{2}(\cosh\left(2y% \right)+\cos\left(2x\right))\right)^{1/2},$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $x$ : real variable, $y$ : real variable and $z$ : complex variable A&S Ref: 4.3.61 Permalink: http://dlmf.nist.gov/4.21.E42 Encodings: TeX, pMML, png See also: Annotations for §4.21(iv), §4.21 and Ch.4

4.21.43		$\|\tan z\|=\left(\frac{\cosh\left(2y\right)-\cos\left(2x\right)}{\cosh\left(2y% \right)+\cos\left(2x\right)}\right)^{1/2}.$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\cosh\NVar{z}$ : hyperbolic cosine function, $\tan\NVar{z}$ : tangent function, $x$ : real variable, $y$ : real variable and $z$ : complex variable A&S Ref: 4.3.63 Permalink: http://dlmf.nist.gov/4.21.E43 Encodings: TeX, pMML, png See also: Annotations for §4.21(iv), §4.21 and Ch.4

4.20 Derivatives and Differential Equations 4.22 Infinite Products and Partial Fractions

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov