§4.17 Special Values and Limits

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Table 4.17.1: Trigonometric functions: values at multiples of

\frac{1}{12}\pi

$\theta$	$\sin\theta$	$\cos\theta$	$\tan\theta$	$\csc\theta$	$\sec\theta$	$\cot\theta$
0	0	1	0	$\infty$	1	$\infty$
$\pi/12$	$\frac{1}{4}\sqrt{2}(\sqrt{3}-1)$	$\frac{1}{4}\sqrt{2}(\sqrt{3}+1)$	$2-\sqrt{3}$	$\sqrt{2}(\sqrt{3}+1)$	$\sqrt{2}(\sqrt{3}-1)$	$2+\sqrt{3}$
$\pi/6$	$\frac{1}{2}$	$\frac{1}{2}\sqrt{3}$	$\frac{1}{3}\sqrt{3}$	$2$	$\frac{2}{3}\sqrt{3}$	$\sqrt{3}$
$\pi/4$	$\frac{1}{2}\sqrt{2}$	$\frac{1}{2}\sqrt{2}$	$1$	$\sqrt{2}$	$\sqrt{2}$	$1$
$\pi/3$	$\frac{1}{2}\sqrt{3}$	$\frac{1}{2}$	$\sqrt{3}$	$\frac{2}{3}\sqrt{3}$	$2$	$\frac{1}{3}\sqrt{3}$
$5\pi/12$	$\frac{1}{4}\sqrt{2}(\sqrt{3}+1)$	$\frac{1}{4}\sqrt{2}(\sqrt{3}-1)$	$2+\sqrt{3}$	$\sqrt{2}(\sqrt{3}-1)$	$\sqrt{2}(\sqrt{3}+1)$	$2-\sqrt{3}$
$\pi/2$	1	0	$\infty$	1	$\infty$	0
$7\pi/12$	$\frac{1}{4}\sqrt{2}(\sqrt{3}+1)$	$-\frac{1}{4}\sqrt{2}(\sqrt{3}-1)$	$-(2+\sqrt{3})$	$\sqrt{2}(\sqrt{3}-1)$	$-\sqrt{2}(\sqrt{3}+1)$	$-(2-\sqrt{3})$
$2\pi/3$	$\frac{1}{2}\sqrt{3}$	$-\frac{1}{2}$	$-\sqrt{3}$	$\frac{2}{3}\sqrt{3}$	$-2$	$-\frac{1}{3}\sqrt{3}$
$3\pi/4$	$\frac{1}{2}\sqrt{2}$	$-\frac{1}{2}\sqrt{2}$	$-1$	$\sqrt{2}$	$-\sqrt{2}$	$-1$
$5\pi/6$	$\frac{1}{2}$	$-\frac{1}{2}\sqrt{3}$	$-\frac{1}{3}\sqrt{3}$	$2$	$-\frac{2}{3}\sqrt{3}$	$-\sqrt{3}$
$11\pi/12$	$\frac{1}{4}\sqrt{2}(\sqrt{3}-1)$	$-\frac{1}{4}\sqrt{2}(\sqrt{3}+1)$	$-(2-\sqrt{3})$	$\sqrt{2}(\sqrt{3}+1)$	$-\sqrt{2}(\sqrt{3}-1)$	$-(2+\sqrt{3})$
$\pi$	0	$-1$	$0$	$\infty$	$-1$	$\infty$

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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 and $\tan\NVar{z}$ : tangent function
Keywords:: special values, trigonometric functions
A&S Ref:: 4.3.46
Permalink:: http://dlmf.nist.gov/4.17.T1
See also:: Annotations for §4.17 and Ch.4

4.17.1	$\displaystyle\lim_{z\to 0}\frac{\sin z}{z}$	$\displaystyle=1,$
ⓘ Symbols: $\sin\NVar{z}$ : sine function and $z$ : complex variable A&S Ref: 4.3.74 Permalink: http://dlmf.nist.gov/4.17.E1 Encodings: TeX, pMML, png See also: Annotations for §4.17 and Ch.4
4.17.2	$\displaystyle\lim_{z\to 0}\frac{\tan z}{z}$	$\displaystyle=1.$
ⓘ Symbols: $\tan\NVar{z}$ : tangent function and $z$ : complex variable A&S Ref: 4.3.75 Permalink: http://dlmf.nist.gov/4.17.E2 Encodings: TeX, pMML, png See also: Annotations for §4.17 and Ch.4
4.17.3	$\displaystyle\lim_{z\to 0}\frac{1-\cos z}{z^{2}}$	$\displaystyle=\frac{1}{2}.$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function and $z$ : complex variable Permalink: http://dlmf.nist.gov/4.17.E3 Encodings: TeX, pMML, png See also: Annotations for §4.17 and Ch.4