Trigonometric function

In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. They are commonly defined as ratios of two sides of a right triangle containing the angle, and can equivalently be defined as the lengths of various line segments from a unit circle. More modern definitions express them as infinite series or as solutions of certain differential equations, allowing their extension to positive and negative values and even to complex numbers. All of these approaches will be presented below.

Properties and applications

The trigonometric functions, as the name suggests, are of crucial importance in trigonometry, mainly because of the following two results:

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Law of sines

The law of sines for an arbitrary triangle states:

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:rac{sin A}{a} = rac{sin B}{b} = rac{sin C}{c}

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It can be proven by dividing the triangle into two right ones and using the above definition of sine. The common number (sinA)/a occurring in the theorem is the reciprocal of the diameter of the circle through the three points A, B and C. The law of sines is useful for computing the lengths of the unknown sides in a triangle if two angles and one side are known. This is a common situation occurring in triangulation, a technique to determine unknown distances by measuring two angles and an accessible enclosed distance.

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Law of cosines

The law of cosines (also known as the cosine formula) is an extension of the Pythagorean theorem:

Related Topics:
Law of cosines - Pythagorean theorem

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:c^2=a^2+b^2-2abcos C ,

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Again, this theorem can be proven by dividing the triangle into two right ones. The law of cosines is useful to determine the unknown data of a triangle if two sides and an angle are known.

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If the angle is not contained between the two sides, the triangle may not be unique. Be aware of this ambiguous case of the Cosine law.

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Law of tangents

There is also a law of tangents:

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:rac{a+b}{a-b} = rac{ an}{ an}

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The trigonometric functions are also important outside of the study of triangles. They are periodic functions with characteristic wave patterns as graphs, useful for modelling recurring phenomena such as sound or light waves. Every signal can be written as a (typically infinite) sum of sine and cosine functions of different frequencies; this is the basic idea of Fourier analysis, where trigonometric series are used to solve a variety of boundary-value problems in partial differential equations.

Related Topics:
Periodic function - Wave - Fourier analysis

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The image on the right displays a two-dimensional graph based on such a summation of sines and cosines, illustrating the fact that arbitrarily complicated closed curves can be described by a Fourier series. Its equation is:

Related Topics:
Closed curve - Fourier series

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: (x( heta),,y( heta)) = sum_{n=1}^infty rac {1}{F(n+1)} (sin( hetacdot F(n)),, cos( hetacdot F(n)))

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where F(n) is the nth Fibonacci number.

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For a compilation of many relations between the trigonometric functions, see trigonometric identities.

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