Secans and coscans

Definitions on the unit circle

Secans and coscans are trigonometric functions . The secant is denoted by, the cosecant by or . The functions get their name from the definition in the unit circle . The function values ​​correspond to the length of secant sections : ${\displaystyle \sec(x)}$ ${\displaystyle \csc(x)}$ ${\displaystyle \operatorname {cosec} (x)}$

${\displaystyle {\overline {OT}}=\sec(b)\qquad \qquad {\overline {OK}}=\csc(b)}$

In the right triangle , the secant is the ratio of the hypotenuse to the cathete and thus the reciprocal value function of the cosine function .

The kosekans is the ratio of the hypotenuse to the opposite cathete and thus the reciprocal function of the sine function :

${\displaystyle \sec(\alpha )={\frac {l_{\rm {Hy}}}{l_{\rm {AK}}}}={\frac {c}{b}}\qquad \qquad \csc(\alpha )={\frac {l_{\rm {Hy}}}{l_{\rm {GK}}}}={\frac {c}{a}}}$

${\displaystyle \sec(x)={\frac {1}{\cos(x)}}\qquad \qquad \csc(x)={\frac {1}{\sin(x)}}}$

properties

Graph

Graph of the secant function

Graph of the coscan function

Domain of definition

Secans:	${\displaystyle -\infty <x<+\infty \quad ;\quad x\neq \left(n+{\frac {1}{2}}\right)\cdot \pi \,;\,n\in \mathbb {Z} }$
Cosecant:	${\displaystyle -\infty <x<+\infty \quad ;\quad x\neq n\cdot \pi \ ;\,n\in \mathbb {Z} }$

Range of values

${\displaystyle -\infty <f(x)\leq -1\quad ;\quad 1\leq f(x)<+\infty }$

periodicity

Period length ${\displaystyle 2\cdot \pi \,:\,f(x+2\pi )=f(x)}$

Symmetries

Secans:	Axial symmetry: ${\displaystyle f(x)=f(-x)}$
Cosecant:	Point symmetry: ${\displaystyle f(-x)=-f(x)}$

Poles

Secans:	${\displaystyle x=\left(n+{\frac {1}{2}}\right)\cdot \pi \,;\,n\in \mathbb {Z} }$
Cosecant:	${\displaystyle x=n\cdot \pi \ ;\quad n\in \mathbb {Z} }$

Extreme points

Secans:	Minima:	${\displaystyle x=2n\cdot \pi \,;\,n\in \mathbb {Z} }$	Maxima:	${\displaystyle x=(2n-1)\cdot \pi \ ;\,n\in \mathbb {Z} }$
Cosecant:	Minima:	${\displaystyle x=\left(2n+{\frac {1}{2}}\right)\cdot \pi \ ;\,n\in \mathbb {Z} }$	Maxima:	${\displaystyle x=\left(2n-{\frac {1}{2}}\right)\cdot \pi \ ;\,n\in \mathbb {Z} }$

zeropoint

Both functions have no zeros.

Asymptotes

Both functions have no horizontal asymptotes.

Jump points

Both functions have no jump points.

Turning points

Both functions have no turning points.

Important functional values

Since secant and cosecant are periodic functions with period (corresponds in degree ), it is sufficient to know the function values ​​of the secant for the area and those of the cosecant for the area . Function values ​​outside this range can therefore be due to the periodicity through the context ${\displaystyle 2\pi }$ ${\displaystyle 360^{\circ }}$ ${\displaystyle 0\leq x\leq 2\pi \,;\quad x\neq {\frac {\pi }{2}},x\neq {\frac {3\pi }{2}}}$ ${\displaystyle 0\leq x\leq 2\pi \,;\quad x\neq 0,x\neq \pi ,x\neq 2\pi }$

${\displaystyle \sec(x)=\sec(x+2k\pi )\quad {\text{und}}\quad \csc(x)=\csc(x+2k\pi )}$

to be determined. In degrees, the relationship is analogous

${\displaystyle \sec(x)=\sec(x+k\cdot 360^{\circ })\quad {\text{und}}\quad \csc(x)=\csc(x+k\cdot 360^{\circ })\,.}$

Here denotes an integer . The following table lists the important function values ​​of the two trigonometric functions in an easy-to-remember series. ${\displaystyle k\in \mathbb {Z} }$

Angle (degree)	Radians	Secans	cosecant
${\displaystyle 0^{\circ }}$	${\displaystyle 0}$	${\displaystyle {\frac {2}{\sqrt {4}}}=1}$	${\displaystyle -}$
${\displaystyle 30^{\circ }}$	${\displaystyle {\frac {\pi }{6}}}$	${\displaystyle {\frac {2}{\sqrt {3}}}={\frac {2}{3}}{\sqrt {3}}}$	${\displaystyle {\frac {2}{\sqrt {1}}}=2}$
${\displaystyle 45^{\circ }}$	${\displaystyle {\frac {\pi }{4}}}$	${\displaystyle {\frac {2}{\sqrt {2}}}={\sqrt {2}}}$	${\displaystyle {\frac {2}{\sqrt {2}}}={\sqrt {2}}}$
${\displaystyle 60^{\circ }}$	${\displaystyle {\frac {\pi }{3}}}$	${\displaystyle {\frac {2}{\sqrt {1}}}=2}$	${\displaystyle {\frac {2}{\sqrt {3}}}={\frac {2}{3}}{\sqrt {3}}}$
${\displaystyle 90^{\circ }}$	${\displaystyle {\frac {\pi }{2}}}$	${\displaystyle -}$	${\displaystyle {\frac {1}{2}}{\sqrt {4}}=1}$

Other important values ​​are:

Angle (degree)	Radians	Secans	cosecant
${\displaystyle 15^{\circ }}$	${\displaystyle {\tfrac {\pi }{12}}}$	${\displaystyle {\sqrt {6}}-{\sqrt {2}}}$	${\displaystyle {\sqrt {6}}+{\sqrt {2}}}$
${\displaystyle 18^{\circ }}$	${\displaystyle {\tfrac {\pi }{10}}}$	${\displaystyle {\tfrac {1}{5}}{\sqrt {50-10{\sqrt {5}}}}}$	${\displaystyle 1+{\sqrt {5}}}$
${\displaystyle 36^{\circ }}$	${\displaystyle {\tfrac {\pi }{5}}}$	${\displaystyle {\sqrt {5}}-1}$	${\displaystyle {\tfrac {1}{5}}{\sqrt {50+10{\sqrt {5}}}}}$
${\displaystyle 54^{\circ }}$	${\displaystyle {\tfrac {3\pi }{10}}}$	${\displaystyle {\tfrac {1}{5}}{\sqrt {50+10{\sqrt {5}}}}}$	${\displaystyle {\sqrt {5}}-1}$
${\displaystyle 72^{\circ }}$	${\displaystyle {\tfrac {2\pi }{5}}}$	${\displaystyle 1+{\sqrt {5}}}$	${\displaystyle {\tfrac {1}{5}}{\sqrt {50-10{\sqrt {5}}}}}$
${\displaystyle 75^{\circ }}$	${\displaystyle {\tfrac {5\pi }{12}}}$	${\displaystyle {\sqrt {6}}+{\sqrt {2}}}$	${\displaystyle {\sqrt {6}}-{\sqrt {2}}}$
${\displaystyle 180^{\circ }}$	${\displaystyle \pi }$	${\displaystyle -1}$	${\displaystyle -}$
${\displaystyle 270^{\circ }}$	${\displaystyle {\frac {3\pi }{2}}}$	${\displaystyle -}$	${\displaystyle -1}$
${\displaystyle 360^{\circ }}$	${\displaystyle 2\pi }$	${\displaystyle 1}$	${\displaystyle -}$

Evidence Sketches:

• ${\displaystyle \sec(45^{\circ })=\csc(45^{\circ })={\sqrt {2}}}$, because the right triangle in the unit circle (with hypotenuse 1) is then isosceles , and according to Pythagoras applies . ${\displaystyle 1^{2}+1^{2}=x^{2}\Rightarrow x={\sqrt {2}}}$
• ${\displaystyle \sec(60^{\circ })=\csc(30^{\circ })=2}$, because the right-angled triangle in the unit circle (with the hypotenuse 1) is mirrored on the -axis and then equilateral (with side length 1), and thus the side length is twice the length of the opposite cathetus . ${\displaystyle x}$
• ${\displaystyle \sec(30^{\circ })=\csc(60^{\circ })={\tfrac {2}{3}}{\sqrt {3}}}$, because for the right-angled triangle in the unit circle (with hypotenuse 1) for the secant according to Pythagoras applies . ${\displaystyle \sin(30^{\circ })={\tfrac {1}{2}}}$ ${\displaystyle ({\tfrac {1}{x}})^{2}+\left({\tfrac {1}{2}}\right)^{2}=1^{2}\ \Rightarrow \ {\tfrac {1}{x^{2}}}={\tfrac {3}{4}}\ \Rightarrow \ x^{2}={\tfrac {4}{3}}\ \Rightarrow \ x={\tfrac {2}{3}}{\sqrt {3}}}$
• ${\displaystyle \sec(72^{\circ })=\csc(18^{\circ })={\frac {1}{{\tfrac {1}{4}}({\sqrt {5}}-1)}}=1+{\sqrt {5}}}$because the inverse of the golden section occurs in the pentagram , where the halved angle in the tips is 18 °.
• ${\displaystyle \sec(36^{\circ })=\csc(54^{\circ })={\frac {1}{{\tfrac {1}{4}}(1+{\sqrt {5}})}}={\sqrt {5}}-1}$, because the golden ratio occurs in the regular pentagon , where the halved interior angle is equal to 54 °.
• ${\displaystyle \sec(75^{\circ })=\csc(15^{\circ })}$and can be derived with the help of the half-angle formulas for sine and cosine. ${\displaystyle \sec(15^{\circ })=\csc(75^{\circ })}$

Further function values ​​that can be represented with square roots

See also: Sine and Cosine: Further function values ​​that can be specified with square roots

Because of secant each of the reciprocal is the cosine and cosecant is the reciprocal of sine, can the function values and if and with square roots represent when the for and is possible. In general, the four basic arithmetic operations and square roots can be used explicitly and precisely when the angle can be constructed with a compass and ruler , especially if it is of the shape ${\displaystyle \sec(x)}$ ${\displaystyle \sec(x)}$ ${\displaystyle \sin(x)}$ ${\displaystyle \cos(x)}$ ${\displaystyle \csc \alpha \;}$ ${\displaystyle \sec \alpha \;}$ ${\displaystyle \alpha \;}$ ${\displaystyle \alpha \;}$

${\displaystyle \alpha =k{\frac {360^{\circ }}{2^{n}p_{1}\dots p_{r}}}}$

is, where , and which are prime numbers for Fermatsche . ${\displaystyle k\in \mathbb {Z} \;}$ ${\displaystyle n\in \mathbb {N} _{0}\;}$ ${\displaystyle p_{i}\;}$ ${\displaystyle i=1,\dots ,r\;}$

Inverse functions

Secans:

On half a period length, e.g. B. the function is reversible ( arc secans ): ${\displaystyle x\in [0,\pi ]}$

${\displaystyle x=\operatorname {arcsec}(y)}$

cosecant

On half a period length, e.g. B. the function is reversible ( arccosecans ): ${\displaystyle x\in \left[-{\frac {\pi }{2}},{\frac {\pi }{2}}\right]}$

${\displaystyle x=\operatorname {arccsc}(y)}$

Series development

Secans:

${\displaystyle \sec(x)=4\pi \,\sum _{k=0}^{\infty }{\frac {(-1)^{k}(2k+1)}{(2k+1)^{2}\pi ^{2}-4x^{2}}}}$

Cosecant:

${\displaystyle \csc(x)={\frac {1}{x}}-2x\,\sum _{k=1}^{\infty }{\frac {(-1)^{k}}{k^{2}\pi ^{2}-x^{2}}}=\sum _{k=-\infty }^{\infty }{\frac {(-1)^{k}\,x}{x^{2}-k^{2}\pi ^{2}}}}$

Derivation

Secans:

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\sec(x)={\frac {\mathrm {d} }{\mathrm {d} x}}{\frac {\mathrm {1} }{\cos(x)}}={\frac {+\sin(x)}{\cos ^{2}(x)}}=+\sec(x)\cdot \tan(x)=+{\frac {\sec ^{2}(x)}{\csc(x)}}}$

cosecant

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\csc(x)={\frac {\mathrm {d} }{\mathrm {d} x}}{\frac {\mathrm {1} }{\sin(x)}}={\frac {-\cos(x)}{\sin ^{2}(x)}}=-\csc(x)\cdot \cot(x)=-{\frac {\csc ^{2}(x)}{\sec(x)}}}$

integral

Secans:

${\displaystyle \int \sec(x)\,\mathrm {d} x=\ln \left|{\frac {1+\sin(x)}{\cos(x)}}\right|+C=\ln {\Big |}\sec(x)+\tan(x){\Big |}+C=\ln \left|\tan \left({\frac {x}{2}}+{\frac {\pi }{4}}\right)\right|+C={\frac {1}{2}}\ln \left({\frac {1+\sin x}{1-\sin x}}\right)+C}$

cosecant

${\displaystyle \int \csc(x)\,\mathrm {d} x=\ln \left|{\frac {\sin(x)}{1+\cos(x)}}\right|+C=\ln \left|\tan \left({\frac {x}{2}}\right)\right|+C}$

Complex argument

${\displaystyle \sec(x+\mathrm {i} \!\cdot \!y)={\frac {2\cos(x)\cosh(y)}{\cos(2x)+\cosh(2y)}}+\mathrm {i} \;{\frac {2\sin(x)\sinh(y)}{\cos(2x)+\cosh(2y)}}}$   With ${\displaystyle x,y\in \mathbb {R} }$

${\displaystyle \csc(x+\mathrm {i} \!\cdot \!y)={\frac {-2\sin(x)\cosh(y)}{\cos(2x)-\cosh(2y)}}+\mathrm {i} \;{\frac {2\cos(x)\sinh(y)}{\cos(2x)-\cosh(2y)}}}$   With ${\displaystyle x,y\in \mathbb {R} }$

Use for numerical calculations - historical significance

Before electronic calculating machines were ubiquitous, tables were used for the trigonometric functions, mostly in printed books. Multiplying by such a function value from a table was more convenient and practical than dividing by such a value (this also applies to non-increasing root values, etc.); If there is a sine or cosine in the denominator in a formula, it is convenient to write the corresponding cosecant or secant values ​​in the numerator instead of these values.

In the age of generally available electronic pocket calculators, this argument is only of historical significance; Secans and cosecans are no longer mentioned in the newer formula collections and are also not implemented as functions (with their own key) in the computers. For this purpose, these functions have simply become superfluous; they solved a problem that no longer exists.

See also

• Sine and cosine
• Tangent and cotangent
• Trigonometric function
• Hyperbolic secant and hyperbolic cosecant
• Trigonometry formula collection

Web links

• Eric W. Weisstein: Secant and Cosecant on MathWorld

Individual evidence

1. ^ Konstantin A. Semendjajew: Taschenbuch der Mathematik . Verlag Harri Deutsch, 2008, ISBN 3-8171-2007-9 , pp. 1220 ( limited preview in Google Book search).
2. ↑ George Hoever: Higher Mathematics compact . Springer Spectrum, Berlin Heidelberg 2014, ISBN 978-3-662-43994-4 ( limited preview in Google book search).
3. ^ Emil Artin : Galois theory. Verlag Harri Deutsch, Zurich 1973, ISBN 3-87144-167-8 , p. 85.