06-05-2014, 04:02 PM
uses of trigonometry
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Amongst the lay public of non-mathematicians and non-scientists, trigonometry is known chiefly for its application to measurement problems, yet is also often used in ways that are far more subtle, such as its place in the theory of music; still other uses are more technical, such as in number theory. The mathematical topics of Fourier series and Fourier transforms rely heavily on knowledge of trigonometric functions and find application in a number of areas, including statistics.
Fourier series
many fields make use of trigonometry in more advanced ways than can be discussed in a single article. Often those involve what are called Fourier series, after the 18th- and 19th-century French mathematician and physicist Joseph Fourier. Fourier series have a surprisingly diverse array of applications in many scientific fields, in particular in all of the phenomena involving seasonal periodicities mentioned above, and in wave motion, and hence in the study of radiation, of acoustics, of seismology, of modulation of radio waves in electronics, and of electric power engineering.
Fourier transforms
A more abstract concept than Fourier series is the idea of Fourier transform. Fourier transforms involve integrals rather than sums, and are used in a similarly diverse array of scientific fields. Many natural laws are expressed by relating rates of change of quantities to the quantities themselves. For example: The rate of change of population is sometimes jointly proportional to (1) the present population and (2) the amount by which the present population falls short of the carrying capacity. This kind of relationship is called a differential equation. If, given this information, one tries to express population as a function of time, one is trying to "solve" the differential equation. Fourier transforms may be used to convert some differential equations to algebraic equations for which methods of solving them are known. Fourier transforms have many uses. In almost any scientific context in which the words spectrum, harmonic, or resonance are encountered, Fourier transforms or Fourier series are nearby.
Statistics, including mathematical psychology
Intelligence quotients are sometimes held to be distributed according to the bell-shaped curve. About 40% of the area under the curve is in the interval from 100 to 120; correspondingly, about 40% of the population scores between 100 and 120 on IQ tests. Nearly 9% of the area under the curve is in the interval from 120 to 140; correspondingly, about 9% of the population scores between 120 and 140 on IQ tests, etc. Similarly many other things are distributed according to the "bell-shaped curve", including measurement errors in many physical measurements. Why the ubiquity of the "bell-shaped curve"? There is a theoretical reason for this, and it involves Fourier transforms and hence trigonometric functions. That is one of a variety of applications of Fourier transforms to statistics.
Trigonometric functions are also applied when statisticians study seasonal periodicities, which are often represented by Fourier series.
A simple experiment with polarized sunglasses
Suppose one gets two pairs of identical polarized sunglasses (unpolarized sunglasses won't work here), and puts the left lens of one pair atop the right lens of the other, both aligned identically. If one pair is slowly rotated, the amount of light that gets through is observed to decrease until the two lenses are at right angles to each other, when no light gets through. When the angle through which the one pair is rotated is θ, what fractions of the light that penetrates when the angle is 0, gets through? Answer: it is cos2 θ. For example, when the angle is 60 degrees, only 1/4 as much light penetrates the series of two lenses as when the angle is 0 degrees, since the cosine of 60 degrees is 1/2.