4 Elementary FunctionsTrigonometric Functions4.15 Graphics4.17 Special Values and Limits

§4.16 Elementary Properties

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Keywords:

elementary properties, trigonometric functions

Notes:

See Hobson (1928, pp. 19–24).

Permalink:

http://dlmf.nist.gov/4.16

See also:

Annotations for Ch.4

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Permalink:

http://dlmf.nist.gov/4.16.F1

Encodings:

pdf, png

See also:

Annotations for §4.16 and Ch.4

Quadrant	$\sin \theta ,\csc \theta$	$\cos \theta ,\sec \theta$	$\tan \theta ,\cot \theta$
I	$+$	$+$	$+$
II	$+$	$-$	$-$
III	$-$	$-$	$+$
IV	$-$	$+$	$-$

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Symbols:

$\csc z$: cosecant function, $\cos z$: cosine function, $\cot z$: cotangent function, $\sec z$: secant function, $\sin z$: sine function and $\tan z$: tangent function

A&S Ref:

4.3.43

Referenced by:

§4.21(iii)

Permalink:

http://dlmf.nist.gov/4.16.T1

See also:

Annotations for §4.16 and Ch.4

$x$	$-\theta$	$\frac{1}{2}\pi \pm \theta$	$\pi \pm \theta$	$\frac{3}{2}\pi \pm \theta$	$2\pi \pm \theta$
$\sin x$	$-\sin \theta$	$\cos \theta$	$\mp \sin \theta$	$-\cos \theta$	$\pm \sin \theta$
$\cos x$	$\cos \theta$	$\mp \sin \theta$	$-\cos \theta$	$\pm \sin \theta$	$\cos \theta$
$\tan x$	$-\tan \theta$	$\mp \cot \theta$	$\pm \tan \theta$	$\mp \cot \theta$	$\pm \tan \theta$
$\csc x$	$-\csc \theta$	$\sec \theta$	$\mp \csc \theta$	$-\sec \theta$	$\pm \csc \theta$
$\sec x$	$\sec \theta$	$\mp \csc \theta$	$-\sec \theta$	$\pm \csc \theta$	$\sec \theta$
$\cot x$	$-\cot \theta$	$\mp \tan \theta$	$\pm \cot \theta$	$\mp \tan \theta$	$\pm \cot \theta$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\cos z$: cosine function, $\cot z$: cotangent function, $\sec z$: secant function, $\sin z$: sine function, $\tan z$: tangent function and $x$: real variable

A&S Ref:

4.3.44

Permalink:

http://dlmf.nist.gov/4.16.T2

See also:

Annotations for §4.16 and Ch.4

	$\sin \theta =a$	$\cos \theta =a$	$\tan \theta =a$	$\csc \theta =a$	$\sec \theta =a$	$\cot \theta =a$
$\sin \theta$	$a$	${(1-a^2)}^{1/2}$	$a{(1+a^2)}^{-1/2}$	$a^{-1}$	$a^{-1}{(a^2-1)}^{1/2}$	${(1+a^2)}^{-1/2}$
$\cos \theta$	${(1-a^2)}^{1/2}$	$a$	${(1+a^2)}^{-1/2}$	$a^{-1}{(a^2-1)}^{1/2}$	$a^{-1}$	$a{(1+a^2)}^{-1/2}$
$\tan \theta$	$a{(1-a^2)}^{-1/2}$	$a^{-1}{(1-a^2)}^{1/2}$	$a$	${(a^2-1)}^{-1/2}$	${(a^2-1)}^{1/2}$	$a^{-1}$
$\csc \theta$	$a^{-1}$	${(1-a^2)}^{-1/2}$	$a^{-1}{(1+a^2)}^{1/2}$	$a$	$a{(a^2-1)}^{-1/2}$	${(1+a^2)}^{1/2}$
$\sec \theta$	${(1-a^2)}^{-1/2}$	$a^{-1}$	${(1+a^2)}^{1/2}$	$a{(a^2-1)}^{-1/2}$	$a$	$a^{-1}{(1+a^2)}^{1/2}$
$\cot \theta$	$a^{-1}{(1-a^2)}^{1/2}$	$a{(1-a^2)}^{-1/2}$	$a^{-1}$	${(a^2-1)}^{1/2}$	${(a^2-1)}^{-1/2}$	$a$

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Symbols:

$\csc z$: cosecant function, $\cos z$: cosine function, $\cot z$: cotangent function, $\sec z$: secant function, $\sin z$: sine function, $\tan z$: tangent function and $a$: real or complex constant

A&S Ref:

4.3.45

Referenced by:

§4.23(vii), §4.45(i)

Permalink:

http://dlmf.nist.gov/4.16.T3

See also:

Annotations for §4.16 and Ch.4

4.15 Graphics4.17 Special Values and Limits

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