Circle and hyperbolic functions

→ Main article : Trigonometric function

→ Main article : Hyperbolic function

Both the angular functions (eg. As sine , cosine ) and the hyperbolic ( hyperbolic sine , hyperbolic cosine , hyperbolic tangent and cotangent hyperbolic ) are mathematical functions , both for real and complex numbers are defined.

This article only covers the sine and cosine functions in detail. The tangent , cotangent , secant and cosecan functions as well as their analog hyperbolic functions are similar to these in their definitions and properties.

Definitions

Both groups of functions can be defined, among other things, by the exponential function or its Taylor series expansion . The similar names (e.g. sine, hyperbolic sine) can be understood through the similar definitions and properties.

Often the circle and hyperbolic functions differ in definition or properties only in that the function variable of the circle function is replaced by the product of the imaginary unit with the function variable, or the positive and negative signs are exchanged.

Definition via the exponential function

The definitions of circle and hyperbolic functions using the exponential function allow the functional behavior to be traced back to a known function. They are therefore used frequently.

${\displaystyle \sin(z)={\frac {e^{iz}-e^{-iz}}{2i}}}$

${\displaystyle \sinh(z)={\frac {e^{z}-e^{-z}}{2}}}$

${\displaystyle \cos(z)={\frac {e^{iz}+e^{-iz}}{2}}}$

${\displaystyle \cosh(z)={\frac {e^{z}+e^{-z}}{2}}}$

Derivation

The notation des and des as the sum of exponential functions can be derived from Euler's formula . Euler's formula is: ${\displaystyle \sin(z)}$ ${\displaystyle \cos(z)}$

${\displaystyle e^{i\varphi }=\cos \varphi +i\sin \varphi }$.

It also follows

${\displaystyle e^{-i\varphi }=\cos \varphi -i\sin \varphi }$.

Since the cosine is an even function , the minus sign can be omitted. The sine is odd and you can therefore put the minus sign in front of the function.

If you now subtract the second equation from the first and solve for, then you get the above equation for . The formula for is obtained analogously. The two equations then have to be added. ${\displaystyle \sin \varphi }$ ${\displaystyle \sin(z)}$ ${\displaystyle \cos(z)}$

Definition of series development

The Taylor series with the development point z = 0 differ only in the signs of every second sum term. In the case of the hyperbolic functions, all the elements in the series are added; in the case of the circular functions, every second row element is subtracted.

${\displaystyle {\begin{aligned}\sin(z)&=z-{\frac {z^{3}}{3!}}+{\frac {z^{5}}{5!}}-{\frac {z^{7}}{7!}}+\dots =\sum _{n=0}^{\infty }(-1)^{n}{\frac {z^{2n+1}}{(2n+1)!}}\\\sinh(z)&=z+{\frac {z^{3}}{3!}}+{\frac {z^{5}}{5!}}+{\frac {z^{7}}{7!}}+\dots =\sum _{n=0}^{\infty }{\frac {z^{2n+1}}{(2n+1)!}}\\\cos(z)&=1-{\frac {z^{2}}{2!}}+{\frac {z^{4}}{4!}}-{\frac {z^{6}}{6!}}+\dots =\sum _{n=0}^{\infty }(-1)^{n}{\frac {z^{2n}}{(2n)!}}\\\cosh(z)&=1+{\frac {z^{2}}{2!}}+{\frac {z^{4}}{4!}}+{\frac {z^{6}}{6!}}+\dots =\sum _{n=0}^{\infty }{\frac {z^{2n}}{(2n)!}}\end{aligned}}}$

Here is the expression n ! for the factorial of n , the product of the first n natural numbers :

${\displaystyle n!=1\cdot 2\cdot \dots \cdot n}$ especially too ${\displaystyle 0!=1}$

Features of the functions

Circle and hyperbola

The name circle or hyperbolic functions comes from the fact that the circle functions describe a circle and the hyperbolic functions describe a hyperbola . Let u be the enclosed area between the x -axis, the graph y ( x ) and the straight line connecting the origin and a point on the graph. For the circular functions, u is also equal to half the angle in radians between the straight line connecting the origin and a point on the graph and the x -axis. For example, a quarter circle, i.e. an area of u = π / 4, corresponds to an angle of π / 2. When the angle function is reversed, an arc is obtained, hence: arcsine and arcsine . Only the definition with the surface applies to the hyperbolic functions. Therefore, the inverse function results in a surface: Areasinus hyperbolicus and Areakosinus hyperbolicus . ${\displaystyle x^{2}+y^{2}=1}$ ${\displaystyle x^{2}-y^{2}=1}$

Circle functions:

${\displaystyle \sin ^{2}(u)+\cos ^{2}(u)=1\quad {\text{mit}}\quad y=\sin(u)\quad {\text{und}}\quad x=\cos(u)}$

${\displaystyle \Longrightarrow \quad {\text{Kreisgleichung}}\quad y=\pm {\sqrt {1-x^{2}}}}$

Hyperbolic functions:

${\displaystyle \cosh ^{2}(u)-\sinh ^{2}(u)=1\quad {\text{mit}}\quad y=\sinh(u)\quad {\text{und}}\quad x=\cosh(u)}$

${\displaystyle \Longrightarrow \quad {\text{Hyperbelgleichung}}\quad y=\pm {\sqrt {x^{2}-1}}}$

Conversion between circular and hyperbolic functions

The following applies to all : ${\displaystyle z\in \mathbb {C} }$

${\displaystyle {\begin{array}{rcr}\sin(i\cdot z)&=&i\cdot \sinh(z)\\\sinh(i\cdot z)&=&i\cdot \sin(z)\\\cos(i\cdot z)&=&\cosh(z)\\\cosh(i\cdot z)&=&\cos(z)\end{array}}}$

respectively:

${\displaystyle {\begin{array}{rcr}\sin(z)&=&-i\cdot \sinh(i\cdot z)\\\sinh(z)&=&-i\cdot \sin(i\cdot z)\\\cos(z)&=&\cosh(i\cdot z)\\\cosh(z)&=&\cos(i\cdot z)\end{array}}}$

The Gudermann function offers another possibility of converting the circle and hyperbolic functions into one another . The advantage is that the detour via the complex numbers can be avoided.

${\displaystyle \sinh(x)=\tan(\operatorname {gd} (x))\ }$

${\displaystyle \cosh(x)=\sec(\operatorname {gd} (x))\ }$

${\displaystyle \tanh(x)=\sin(\operatorname {gd} (x))\ }$

${\displaystyle \operatorname {sech} (x)=\cos(\operatorname {gd} (x))\ }$

${\displaystyle \operatorname {csch} (x)=\cot(\operatorname {gd} (x))\ }$

${\displaystyle \coth(x)=\csc(\operatorname {gd} (x))\ }$

Derivatives

The derivatives of the circular and hyperbolic functions are also similar to one another.

${\displaystyle {\begin{matrix}\sin '(z)&=&\cos(z)\\\sinh '(z)&=&\cosh(z)\\\cos '(z)&=&-\sin(z)\\\cosh '(z)&=&\sinh(z)\end{matrix}}}$

Addition theorems (goniometric relationships)

The following addition theorems apply to both the circular and the hyperbolic functions:

${\displaystyle {\begin{matrix}\sin(x\pm y)&=&\sin(x)\cos(y)\pm \cos(x)\sin(y)\\\cos(x\pm y)&=&\cos(x)\cos(y)\mp \sin(x)\sin(y)\\\tan(x\pm y)&=&\displaystyle {\frac {\tan(x)\pm \tan(y)}{1\mp \tan(x)\tan(y)}}\\\\\end{matrix}}}$

${\displaystyle \sinh(x\pm y)=\sinh(x)\cosh(y)\pm \cosh(x)\sinh(y)}$

${\displaystyle \cosh(x\pm y)=\cosh(x)\cosh(y)\pm \sinh(x)\sinh(y)}$

${\displaystyle \tanh(x\pm y)={\frac {\tanh(x)\pm \tanh(y)}{1\pm \tanh(x)\tanh(y)}}}$

${\displaystyle \sin(x)+\sin(y)=2\sin \left({\frac {x+y}{2}}\right)\cdot \cos \left({\frac {x-y}{2}}\right)}$

${\displaystyle \sin(x)-\sin(y)=2\cos \left({\frac {x+y}{2}}\right)\cdot \sin \left({\frac {x-y}{2}}\right)}$

${\displaystyle \cos(x)+\cos(y)=2\cos \left({\frac {x+y}{2}}\right)\cdot \cos \left({\frac {x-y}{2}}\right)}$

${\displaystyle \cos(x)-\cos(y)=-2\sin \left({\frac {x+y}{2}}\right)\cdot \sin \left({\frac {x-y}{2}}\right)}$

${\displaystyle \sinh(x)+\sinh(y)=2\sinh \left({\frac {x+y}{2}}\right)\cdot \cosh \left({\frac {x-y}{2}}\right)}$

${\displaystyle \sinh(x)-\sinh(y)=2\cosh \left({\frac {x+y}{2}}\right)\cdot \sinh \left({\frac {x-y}{2}}\right)}$

${\displaystyle \cosh(x)+\cosh(y)=2\cosh \left({\frac {x+y}{2}}\right)\cdot \cosh \left({\frac {x-y}{2}}\right)}$

${\displaystyle \cosh(x)-\cosh(y)=2\sinh \left({\frac {x+y}{2}}\right)\cdot \sinh \left({\frac {x-y}{2}}\right)}$

${\displaystyle \sin(x)\cdot \sin(y)={\frac {1}{2}}{\Big [}\cos(x-y)-\cos(x+y){\Big ]}}$

${\displaystyle \cos(x)\cdot \cos(y)={\frac {1}{2}}{\Big [}\cos(x-y)+\cos(x+y){\Big ]}}$

${\displaystyle \sin(x)\cdot \cos(y)={\frac {1}{2}}{\Big [}\sin(x-y)+\sin(x+y){\Big ]}}$

${\displaystyle \sinh(x)\cdot \sinh(y)={\frac {1}{2}}{\Big [}\cosh(x+y)-\cosh(x-y){\Big ]}}$

${\displaystyle \cosh(x)\cdot \cosh(y)={\frac {1}{2}}{\Big [}\cosh(x+y)+\cosh(x-y){\Big ]}}$

${\displaystyle \sinh(x)\cdot \cosh(y)={\frac {1}{2}}{\Big [}\sinh(x+y)+\sinh(x-y){\Big ]}}$

For further relationships see also the trigonometry formula collection .

swell

1. ↑ dtv atlas of mathematics . Volume 1. Deutscher Taschenbuch Verlag, Munich, 4th edition 1980. ISBN 3-423-03007-0 . P. 185.
2. ↑ Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions , Dover Publications, New York 1964, ISBN 0-486-61272-4 , Chapters 4.3 and 4.5