4 Elementary Functions Trigonometric Functions 4.15 Graphics 4.17 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

See accompanying text — Figure 4.16.1: Quadrants for the angle $\theta$ . Magnify

Table 4.16.1: Signs of the trigonometric functions in the four quadrants.

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

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Symbols:: $\csc\NVar{z}$ : cosecant function, $\cos\NVar{z}$ : cosine function, $\cot\NVar{z}$ : cotangent function, $\sec\NVar{z}$ : secant function, $\sin\NVar{z}$ : sine function and $\tan\NVar{z}$ : 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

Table 4.16.2: Trigonometric functions: quarter periods and change of sign.

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

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\csc\NVar{z}$ : cosecant function, $\cos\NVar{z}$ : cosine function, $\cot\NVar{z}$ : cotangent function, $\sec\NVar{z}$ : secant function, $\sin\NVar{z}$ : sine function, $\tan\NVar{z}$ : 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

Table 4.16.3: Trigonometric functions: interrelations. All square roots have their principal values when the functions are real, nonnegative, and finite.

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

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Symbols:: $\csc\NVar{z}$ : cosecant function, $\cos\NVar{z}$ : cosine function, $\cot\NVar{z}$ : cotangent function, $\sec\NVar{z}$ : secant function, $\sin\NVar{z}$ : sine function, $\tan\NVar{z}$ : 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