Trigonometric functions

With trigonometric functions or trigonometric functions is called arithmetical relationships between angles and aspect ratios (originally in right triangles ). Tables with ratio values ​​for certain angles allow calculations for surveying tasks that use angle and side lengths in triangles. The trigonometric functions are also the basic features for the description of periodic processes in the natural sciences.

Overview of the trigonometric functions

The elementary trigonometric functions are:

• The sine function (abbreviated sin),
• The cosine (abbreviated cos )
• The tangent function (abbreviated tan or tg)

And their inverse functions

• Kosekansfunktion ( reciprocal of the sine: csc )
• Sekansfunktion ( reciprocal of the cosine: sec )
• Cotangent ( inverse of the tangent: cot )

There are close relationships between these functions. In fact, one of the functions would be sufficient to solve any trigonometric problems. However, the use of several different functions enables simplification of the calculations and formulas.

The cotangent is often used in tables of function values ​​of trigonometric functions, since one can tabulate cot (x ) together with the tangent function. So far as the importance of cot (x ) is somewhat greater than that of s ( x) and CSC (x).

There are more - today rather unusual - features such as sinus versus ( versin ) versus cosine ( coversin ) exsecant ( exsec ) and excosecant ( excsc ).

Definition

Originally, the trigonometric functions are defined as aspect ratios in right triangles and therefore only for angles of 0 to 90 degrees:

This definition is independent of the choice of the right triangle which is used in the calculation. In any right triangle with the same angle these relations yield the same value. This can for example be prove the beam sets.

From these relations, the relationship directly follows:

The adjacent side of the angle is also the opposite side of the other acute angle of the right triangle; as the sum of the angles in a triangle is 180 °, and the right angle is 90 ° contributing to this sum, this angle and therefore

However, the trigonometric functions can be expanded as secant and tangent sections of the unit circle to larger angles. From the intersection of an angle leg to the unit circle, the solders are like on the two coordinate axes and provide sine and cosine of the angle. The tangents at the points x = 1 and y = 1 cut the legs also and then deliver in the projection on the axes of the tangent and the cotangent. Here, the leg may need to be extended rearwardly, in order to obtain an intersection point. In this way, any angle from 0 to 360 degrees values ​​of the trigonometric functions can be assigned, the course can now be negative (see figure). The relations given above apply while continuing.

In calculus, the sine and cosine are usually defined via power series, where the angle is expressed in radians. For details, see in Articles sine and cosine, and tangent.

Relations between the functions

The signs of the trigonometric functions depending on the quadrant indicates the following table:

The amount will be converted as follows:

Application of the trigonometric functions

Mainly, the trigonometric functions are used in surveying. For a list of formulas for the calculation of the triangle sizes see the article in triangle geometry.

Furthermore, they are important in the analysis, and in many applications the physics. There is a close relationship to the exponential function, which is particularly evident in complex numbers and functions in the Taylor series of the functions.

Reversal of the trigonometric functions

In some situations, the trigonometric functions of angles are needed to calculate aspect ratios of angle. For this, the Inverse Trigonometric Functions or inverse trigonometric functions are arcsin, arccos, arctan and arccot ​​- the inverse functions of the trigonometric functions - used. On calculators, they are often referred to as sin-1 etc.. This is consistent with the notation for the inverse function of f coincide (even if the Inverse Trigonometric Functions which strictly speaking are not ), but collides with the equally usual convention for writing.

Inverse Trigonometric Functions that are used to calculate an aspect ratio for the angle. Because of the symmetry of the trigonometric functions should be clarified on a case by case in which quadrant the angle is searched.