The trigonometric functions are also important in physics. The sine and the cosine functions, for example, are used to describe simple harmonic motion, which models many natural phenomena, such as the movement of a mass attached to a spring and, for small angles, the pendular motion of a mass hanging by a string. The sine and cosine functions are one-dimensional projections of uniform circular motion.
Trigonometric functions also prove to be useful in the study of general periodic functions. The characteristic wave patterns of periodic functions are useful for modeling recurring phenomena such as sound or light waves.
Under rather general conditions, a periodic function ƒ(x) can be expressed as a sum of sine waves or cosine waves in a Fourier series. Denoting the sine or cosine basis functions by φk, the expansion of the periodic function ƒ(t) takes the form:
For example, the square wave can be written as the Fourier series
In the animation of a square wave at top right it can be seen that just a few terms already produce a fairly good approximation. The superposition of several terms in the expansion of a sawtooth wave are shown underneath.
Other articles related to "periodic functions, functions":
... The space Wp of Weyl almost periodic functions (for p ≥ 1) was introduced by Weyl (1927) ... It contains the space Sp of Stepanov almost periodic functions ... It is the closure of the trigonometric polynomials under the seminorm Warning there are nonzero functions ƒ with
1921) was a Latvian mathematician, who worked in differential equations, topology and quasi-periodic functions ... This was for an investigation of quasi-periodic functions ... The notion of quasi-periodic functions was generalised still further by Harald Bohr when he introduced almost-periodic functions ...
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