4 Elementary FunctionsTrigonometric Functions4.20 Derivatives and Differential Equations4.22 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

Contents

1. §4.21(i) Addition Formulas
2. §4.21(ii) Squares and Products
3. §4.21(iii) Multiples of the Argument
4. §4.21(iv) Real and Imaginary Parts; Moduli

§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 $\nu =\frac{1}{4}\pi$.

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		$$\sin u\pm \cos u=\sqrt{2}\sin \left(u\pm \frac{1}{4}\pi \right)=\pm \sqrt{2}\cos \left(u\mp \frac{1}{4}\pi \right).$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function and $\sin z$: 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		$$A\cos u+B\sin u=\sqrt{A^2+B^2}\cos \left(u-\mathrm{ph}\left(A+B\mathrm{i}\right)\right),$$
$A,B\in ℝ$,
ⓘ Symbols: $\cos z$: cosine function, $\in$: element of, $\mathrm{i}$: imaginary unit, $\mathrm{ph}$: phase, $ℝ$: real line and $\sin z$: 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		$\sin \left(u\pm v\right)$	$=\sin u\cos v\pm \cos u\sin v,$
ⓘ Symbols: $\cos z$: cosine function and $\sin z$: 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		$\cos \left(u\pm v\right)$	$=\cos u\cos v\mp \sin u\sin v,$
ⓘ Symbols: $\cos z$: cosine function and $\sin z$: 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		$\tan \left(u\pm v\right)$	$={\displaystyle \frac{\tan u\pm \tan v}{1\mp \tan u\tan v}},$
ⓘ Symbols: $\tan z$: 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		$\cot \left(u\pm v\right)$	$={\displaystyle \frac{\pm \cot u\cot v-1}{\cot u\pm \cot v}}.$
ⓘ Symbols: $\cot z$: 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		$\sin u+\sin v$	$=2\sin \left({\displaystyle \frac{u+v}{2}}\right)\cos \left({\displaystyle \frac{u-v}{2}}\right),$
ⓘ Symbols: $\cos z$: cosine function and $\sin z$: 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		$\sin u-\sin v$	$=2\cos \left({\displaystyle \frac{u+v}{2}}\right)\sin \left({\displaystyle \frac{u-v}{2}}\right),$
ⓘ Symbols: $\cos z$: cosine function and $\sin z$: 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		$\cos u+\cos v$	$=2\cos \left({\displaystyle \frac{u+v}{2}}\right)\cos \left({\displaystyle \frac{u-v}{2}}\right),$
ⓘ Symbols: $\cos z$: 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		$\cos u-\cos v$	$=-2\sin \left({\displaystyle \frac{u+v}{2}}\right)\sin \left({\displaystyle \frac{u-v}{2}}\right).$
ⓘ Symbols: $\cos z$: cosine function and $\sin z$: 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		$\tan u\pm \tan v$	$={\displaystyle \frac{\sin \left(u\pm v\right)}{\cos u\cos v}},$
ⓘ Symbols: $\cos z$: cosine function, $\sin z$: sine function and $\tan z$: 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		$\cot u\pm \cot v$	$={\displaystyle \frac{\sin \left(v\pm u\right)}{\sin u\sin v}}.$
ⓘ Symbols: $\cot z$: cotangent function and $\sin z$: 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: $\cos z$: cosine function, $\sin z$: 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: $\sec z$: secant function, $\tan z$: 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: $\csc z$: cosecant function, $\cot z$: 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: $\cos z$: cosine function and $\sin z$: 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: $\cos z$: 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: $\cos z$: cosine function and $\sin z$: 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		${\sin }^{2}u-{\sin }^{2}v$	$=\sin \left(u+v\right)\sin \left(u-v\right),$
ⓘ Symbols: $\sin z$: 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		${\cos }^{2}u-{\cos }^{2}v$	$=-\sin \left(u+v\right)\sin \left(u-v\right),$
ⓘ Symbols: $\cos z$: cosine function and $\sin z$: 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		${\cos }^{2}u-{\sin }^{2}v$	$=\cos \left(u+v\right)\cos \left(u-v\right).$
ⓘ Symbols: $\cos z$: cosine function and $\sin z$: 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: $\cos z$: cosine function, $\sin z$: 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: $\cos z$: 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: $\cos z$: cosine function, $\sin z$: sine function, $\tan z$: 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		$\sin \left(-z\right)$	$=-\sin z,$
ⓘ Symbols: $\sin z$: 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		$\cos \left(-z\right)$	$=\cos z,$
ⓘ Symbols: $\cos z$: 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		$\tan \left(-z\right)$	$=-\tan z.$
ⓘ Symbols: $\tan z$: 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: $\cos z$: cosine function, $\sin z$: sine function, $\tan z$: 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: $\cos z$: cosine function, $\sin z$: sine function, $\tan z$: 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: $\cot z$: cotangent function, $\tan z$: 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		$\sin \left(3z\right)$	$=3\sin z-4{\sin }^{3}z,$
ⓘ Symbols: $\sin z$: 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		$\cos \left(3z\right)$	$=-3\cos z+4{\cos }^{3}z,$
ⓘ Symbols: $\cos z$: 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		$\sin \left(4z\right)$	$=8{\cos }^{3}z\sin z-4\cos z\sin z,$
ⓘ Symbols: $\cos z$: cosine function, $\sin z$: 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		$\cos \left(4z\right)$	$=8{\cos }^{4}z-8{\cos }^{2}z+1.$
ⓘ Symbols: $\cos z$: 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		$$\cos \left(nz\right)+\mathrm{i}\sin \left(nz\right)={(\cos z+\mathrm{i}\sin z)}^n.$$
ⓘ Symbols: $\cos z$: cosine function, $\mathrm{i}$: imaginary unit, $\sin z$: 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 $-\pi \le \mathrm{\Re }z\le \pi$.

4.21.35		$$\sin \left(nz\right)=2^{n-1}\prod _{k=0}^{n-1}\sin \left(z+\frac{k\pi }{n}\right),$$
$n=1,2,3,\mathrm{\dots }$.
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: 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		$\sin z$	$={\displaystyle \frac{2t}{1+t^2}},$
		$\cos z$	$={\displaystyle \frac{1-t^2}{1+t^2}},$
		$dz$	$={\displaystyle \frac{2}{1+t^2}}dt.$
ⓘ Symbols: $\cos z$: cosine function, $dx$: differential of $x$, $\sin z$: 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 $z=x+\mathrm{i}y$

4.21.37		$$\sin z=\sin x\cosh y+\mathrm{i}\cos x\sinh y,$$
ⓘ Symbols: $\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\sin z$: 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: $\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\sin z$: 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: $\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $\tan z$: 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: $\cos z$: cosine function, $\cot z$: cotangent function, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\sin z$: 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		$$|\sin z|={({\sin }^{2}x+{\sinh }^{2}y)}^{1/2}={\left(\frac{1}{2}\left(\cosh \left(2y\right)-\cos \left(2x\right)\right)\right)}^{1/2},$$
ⓘ Symbols: $\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\sin z$: 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		$$|\cos z|={({\cos }^{2}x+{\sinh }^{2}y)}^{1/2}={\left(\frac{1}{2}(\cosh \left(2y\right)+\cos \left(2x\right))\right)}^{1/2},$$
ⓘ Symbols: $\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\sinh z$: 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: $\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\tan z$: 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 Equations4.22 Infinite Products and Partial Fractions

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