Trigonometry (from the Greek τρίγωνον trigonon, triangle ' and μέτρον métron, measure ') is a branch of geometry and therefore mathematics. As far as problems of plane geometry ( plane geometry ) are treated trigonometry, it is called planar trigonometry; in addition there are the spherical trigonometry, which deals with ball triangles ( spherical triangles ), and the hyperbolic trigonometry. The following comments relate primarily to the field of plane trigonometry.
The basic task of trigonometry is ( etc. page lengths, angles, sizes, lengths of Dreieckstransversalen ) calculated from three sizes of a given triangle other sizes of this triangle. ( Csc ) as an aid to the trigonometric functions ( trigonometric functions, circular functions, goniometric functions ) Sine (sin ), cosine ( cos), tangent (tan ), Cotangent (cot), secant (sec) and cosecant used. Trigonometric calculations can also refer to more complex geometric objects such as polygons ( polygons ), on problems of solid geometry ( solid geometry ) and on questions of many other countries ( see below).
Trigonometry in right-angled triangle
Particularly simple is the trigonometry of the right triangle. Since the sum of the angles of a triangle is 180 degrees, the right angle of such a triangle is the largest interior angle. He is the longest side ( called the hypotenuse ) against. The two shorter sides of the triangle is called catheti. If one refers to one of the two smaller angle, it is useful to distinguish between ( adjacent to the given angle ) between the opposite side ( the given angle opposite) and the adjacent side. We define now:
It is not a matter of course that these definitions make sense. Of the observed triangle namely only the sizes of the angles are known, but not the side lengths. But different right-angled triangles with the given angles are at least similar to each other, so that they are consistent in their aspect ratios. For example, one of these triangles is twice as long as the other sites have. The fractions of the above defining equations have the same values in this case. Thus, these values depend only on the given angle. For this reason it makes sense to speak of functions of the angle.
Example: Calculation of side
The following numerical values are rounded. In a triangle ABC the following quantities are given:
From this information, the page length c is to be determined. Since the adjacent side to the hypotenuse of known and is sought, the cosine function is used.
Example: Calculation of angular size
From a triangle ABC is known:
Wanted is the angle. The two given sides and the adjacent side and the opposite side of. Therefore, it is sensible to use the tangent function.
During the last example was to be calculated for a known angle of the cosine, here the situation is reversed. For a known value of the associated tangent angle to be determined. This will require the inverse of the tangent function, so-called arc tangent function ( arctan ). With this we obtain:
Definition of trigonometric functions on the unit circle
The definitions used so far are useful only for angles below 90 °. For many purposes, but one larger angle is interested in trigonometric values. The unit circle is a circle with radius 1, permits such an extension of the previous definition. At the appropriate angle of the corresponding point on the unit circle is determined. The x-coordinate of that point is the cosine of the given angle, the y- coordinate of the sine value.
As defined above of the sine and cosine value of the x- and y-coordinates can be easily extended to more than 90 angular degrees. It can be seen here that for angles between 90 ° and 270 ° the x-coordinate, and thus the cosine is negative, corresponding to the angle between 180 ° and 360 °, the y- coordinate, and hence the sinus. Also angles larger than 360 °, and negative angle can be transmitted to define easily.
It should be noted that in the modern approach, the relationship between sine and cosine of angle and is used in order to define the angle. The sine and cosine function itself are introduced via their series representation.
The other four trigonometric functions are defined by:
- Graphs of the main trigonometric functions (angle in radians, ie π ≙ 180 ° )
Trigonometry in general triangle
Also for general triangles several formulas have been developed that make it possible to determine unknown side lengths or angle sizes. One can think here in particular of the law of sines and cosines of. The use of the sine theorem
Is useful are known, either of two sides of a triangle and one of the two angles opposite or one side and two angles. The law of cosines
Allows either to calculate the angle of the given three sides or from two sides and its intermediate angle, the opposite side. Other formulas that are valid for arbitrary triangles, are the tangent set, the half-angle set ( Kotangenssatz ) and the minor pasture between formulas.
Properties and formulas
The articles about the six trigonometric functions (sine, cosine, tangent, cotangent, secant, Kosecans ) and the formulary trigonometry contain numerous properties of these functions and formulas for computing with them. Particularly frequently used complementary formulas for sine and cosine
And the trigonometric Pythagoras
Also important are the addition theorems of trigonometric functions and the consequences of it. It involves trigonometric values of sums or differences of angles. So true example for all and:
Other identities can be found in the formulary trigonometry.
Areas of application
Trigonometry plays a crucial role in many areas:
In geodesy (surveying ) is called triangulation, when one of points of known position of other points anpeilt (angle measurement) and from trigonometry determines the positions of the new points. In astronomy can be determined in a similar manner the distances of planets, moons and nearby fixed stars. Similarly great is the importance of trigonometry for the navigation of aircraft and ships and for the spherical astronomy, in particular for the calculation of stellar and planetary positions.
In physics, the sine and cosine function used to describe vibrations and waves mathematically. The same applies to the time course of electrical voltage and electrical current in AC technology.
Precursor of trigonometry there was already during Antiquity in Greek mathematics. Aristarchus of Samos took advantage of the properties of right triangles to calculate the distance relationships between the earth and sun and moon. Of the astronomers Hipparchus and Ptolemy is known that they worked with sinew panels, ie, with tables for the conversion of central angles ( central angles ) in chord lengths and vice versa. The values of such tables are directly related to the sine function together: the length of a chord is derived from the radius r and the center angle according to
Similar tables were also used in Indian mathematics. Arab scientists took the results of the Greeks and the Indians and built the trigonometry, particularly spherical trigonometry further. In medieval Europe, the findings of the Arab trigonometry were known until late. The first systematic presentation of the area took place in the 15th century. During the Renaissance, the increasing problems of ballistics and the deep sea shipping required an improvement of trigonometry and of trigonometric panel plant. The German astronomer and mathematician Regiomontanus (Johann Müller ) summarized theorems and methods of plane and spherical trigonometry in the five -volume work De triangulis omnimodis together.
The term trigonometry was Bartholomew Pitiscus in his Trigonometria: introduced sive de solutione triangulorum tractatus brevis et perspicuus of 1595.
The spellings used today and the analytical representation of the trigonometric functions are derived mostly from Leonhard Euler.