Scientific fields that make use of trigonometry include: :
acoustics,
architecture,
astronomy,
cartography,
civil engineering,
geophysics,
crystallography,
electrical engineering,
electronics, land
surveying and
geodesy, many
physical sciences,
mechanical engineering,
machining,
medical imaging,
number theory,
oceanography,
optics,
pharmacology,
probability theory,
seismology,
statistics, and
visual perception That these fields involve trigonometry does not mean knowledge of trigonometry is needed in order to learn anything about them. It
does mean that
some things in these fields cannot be understood without trigonometry. For example, a professor of
music may perhaps know nothing of mathematics, but would probably know that
Pythagoras was the earliest known contributor to the mathematical theory of music. In
some of the fields of endeavor listed above it is easy to imagine how trigonometry could be used. For example, in navigation and land surveying, the occasions for the use of trigonometry are in at least some cases simple enough that they can be described in a beginning trigonometry textbook. In the case of music theory, the application of trigonometry is related to work begun by Pythagoras, who observed that the sounds made by plucking two strings of different lengths are consonant if both lengths are small integer multiples of a common length. The resemblance between the shape of a vibrating string and the graph of the
sine function is no mere coincidence. In oceanography, the resemblance between the shapes of some
waves and the graph of the sine function is also not coincidental. In some other fields, among them
climatology, biology, and economics, there are seasonal periodicities. The study of these often involves the periodic nature of the sine and cosine functions.
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 the
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. A Fourier series is a function
series (infinite sum of functions) of the form : \begin{align} & a_0 + a_1 \cos(\theta) + b_1\sin(\theta) + a_2 \cos (2\theta) + b_2\sin (2\theta) + a_3 \cos(3\theta) + b_3 \cos(3\theta) \dotsb \\ &\qquad = a_0 + \sum_{n=1}^{\infty}\bigl(a_n\cos (n\theta) + b_n\sin (n\theta)\big), \end{align} with real- or complex-valued coefficients and . When the series
converges, it represents a
periodic function of . Fourier used these for studying
heat flow and
diffusion. Fourier series are also applicable to subjects whose connection with wave motion is far from obvious. One ubiquitous example is
digital compression whereby
images,
audio and
video data are compressed into a much smaller size which makes their transmission feasible over
telephone,
internet and
broadcast networks. Another example, mentioned above, is diffusion. Among others are: the
geometry of numbers,
isoperimetric problems, recurrence of
random walks,
quadratic reciprocity, the
central limit theorem,
Heisenberg's inequality.
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 population change 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. 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 a
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. The theoretical reason for the ubiquity of the "bell-shaped curve" 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.
Number theory There is a hint of a connection between trigonometry and number theory. Loosely speaking, one could say that number theory deals with qualitative properties rather than quantitative properties of numbers. : \frac{1}{42}, \qquad \frac{2}{42}, \qquad \frac{3}{42}, \qquad \dots\dots, \qquad \frac{39}{42}, \qquad \frac{40}{42}, \qquad \frac{41}{42}. Discard the ones that are not in lowest terms; keep only those that are in lowest terms: : \frac{1}{42}, \qquad \frac{5}{42}, \qquad \frac{11}{42}, \qquad \dots, \qquad \frac{31}{42}, \qquad \frac{37}{42}, \qquad \frac{41}{42}. Then bring in trigonometry: : \cos\left(2\pi\cdot\frac{1}{42}\right)+ \cos\left(2\pi\cdot\frac{5}{42}\right)+ \cdots+ \cos\left(2\pi\cdot\frac{37}{42}\right)+ \cos\left(2\pi\cdot\frac{41}{42}\right) The value of the sum is −1, because 42 has an
odd number of prime factors and none of them is repeated: 42 = 2 × 3 × 7. (If there had been an
even number of non-repeated factors then the sum would have been 1; if there had been any repeated prime factors (e.g., 60 = 2 × 2 × 3 × 5) then the sum would have been 0; the sum is the
Möbius function evaluated at 42.) This hints at the uses of applying
Fourier analysis to number theory.
Solving non-trigonometric equations Various types of
equations can be solved using trigonometry; for example, a
linear difference equation or
linear differential equation with constant coefficients has solutions expressed in terms of the
eigenvalues of its characteristic equation; if some of the eigenvalues are
complex, the complex terms can be replaced by trigonometric functions of real terms, showing that the dynamic variable exhibits
oscillations. Similarly,
cubic equations with three real solutions have an
algebraic solution that is unhelpful in that it contains cube roots of complex numbers; again an alternative solution exists in terms of trigonometric functions of real terms. ==References==