Tangent function
Tangent and cotangent are trigonometric functions and play a prominent role in mathematics and its applications. The tangent of the angle is denoted by, the cotangent of the angle with. In older literature, one finds also the spellings for the tangent and the cotangent for.
- 2.1 periodicity
- 2.2 monotony
- 2.3 symmetries
- 2.4 Zeroing
- 2.5 pole
- 2.6 turnarounds
Definition
Historical / geometric
The term " tangent " comes from the born in Flensburg mathematician Thomas Finck (1561-1656), who introduced it in 1583. The term " cotangent " evolved from complementi tangent, ie tangent of the complementary angle.
The choice of the name tangent can be explained directly by the definition in the unit circle. The function values correspond to the length of a tangent section:
In a right triangle, the tangent of an angle, the aspect ratio of side to base and the cotangent of the aspect ratio of the adjacent side to opposite side:
It immediately follows:
As well as
Formal - with definition and range of values
Formally, the tangent function by means of the sine and cosine
Be defined, where the range of values are the real or complex numbers depending on the application. In order to prevent that the denominator is zero, the zero points of the cosine function may be omitted in the definition range:
In the real or
In the complexes.
The cotangent can analogously by this
Be defined, where for whose domain
In the real or
In the complex arises when you want to ensure that the denominator is not zero.
For the common domain of definition of and
Applies
Properties
Periodicity
Monotony
Tangent: In each interval strictly increasing.
Cotangent: In each interval strictly decreasing.
Symmetries
Point symmetric about the coordinate origin:
Zeros
Poles
Turning points
Both the tangent function and the cotangent have asymptotes, but no discontinuities or extrema.
Important function values
Inverse function
By suitable restriction of the domains we obtain a bijection
Its inverse function
Ie arctangent and is therefore also bijective.
Its inverse function
Called inverse cotangent and is therefore also bijective.
Series expansion
In this case, designated by the Bernoulli numbers.
The partial fraction expansion of the cotangent is for
Derivation
In the derivation of tangent and cotangent of the otherwise rather uncommon and trigonometric functions secant and cosecant appear:
The - th derivatives can be expressed with the Polygammafunktion:
Primitives
Complex argument
Addition theorems
Are the addition formulas for tangent and cotangent
A symmetric formulation is: If and only applies
Or
If a multiple of is.
From the addition theorems follows, in particular for double angle
( Co- ) tangent representation of sine and cosine
The sine and cosine function can be expressed by tangent or Kotangensfunktionen:
This can be reproduced using the following calculation:
It was exploited. Solving for and delivers above relations.
Rational parametrization
The tangent of the half angle can be used to describe various trigonometric functions by rational expressions: If, as is
It is particularly
A parametrization of the unit circle with the exception of the point ( of the parameter corresponds to ). A parameter value corresponds to the second intersection point of the straight line connecting to and using the unit circle.
Application: tangent and slope angle
The tangent provides an important measure of linear functions: Each linear function has a graph is a straight line. The tangent of the angle between the straight line and the x-axis corresponds exactly to the slope of the line, i.e..
With a negative slope () applies.
The specified as the slope of a road percentage is the tangent of the slope angle.
Applications in physics
Tangent and cotangent can be used to describe the time dependence of the velocity at the throw of a body upwards when the flow resistance of the air, a turbulent flow is set ( Newton friction ). The coordinate system will set so that the local axis is pointing upwards. Then for the speed differential equation in the form of the gravitational acceleration g and a constant k > 0 then the result is:
Wherein the threshold speed is achieved for the case of air resistance. Because of the above strength of relationships between cotangent and tangent can this temporal dependence also just as well be expressed by means of the tangent:
This solution is valid until the body has reached the highest point of its path (ie, when v = 0, that is for ), then it is necessary to use the hyperbolic tangent to describe the following case with air resistance.
Differential equation
The tangent is a solution of the Riccati equation
Factored to the right side, we obtain
With the imaginary unit. The tangent ( as a complex function ), the exception values : These values are not accepted because the constant functions and are solutions of the differential equation and the existence and uniqueness theorem excludes that two different solutions to the same place have the same value.