3.2 Trigonometric functions 63
triangles,
1and is important in structural and architectural design, astronomy, and
navigation. The trigonometric functions are important in the physical sciences for the
description of periodic motion, including circular motion, as in figure 3.1(i), and
wave motion, Figure 3.1(ii). They are also essential for the description of systems with
periodic structure, as in the use of Bragg’s law, equation (1.2) in Chapter 1, for the
interpretation of the diffraction of X-rays from the surfaces of crystal lattices.
The exponential function e
xhas the unique and definitive property that the slope of
its graph at any point (its derivative, see Chapter 4) is equal to the value of the func-
tion at that point. This makes it particularly useful for the modeling of all types of first-
order rate processes, in which the rate of change (growth or decay) of a property is
proportional to the value of the property. Figure 3.1(iii) show exponential growth (with
kpositive) which provides a model of, for example, population explosion, bacterial
growth in a culture, and a nuclear chain reaction. When kis negative, exponentialdecay
provides a model of first-order chemical reactions, radioactive decomposition of nuclei
and, more trivially, the popularity of fashions. In quantum theory, the function e
−ris
the 1sorbital of the hydrogen atom if ris the distance of the electron from the nucleus.
The logarithmic function ln 1 xis the inverse function of e
x. It is often used as an
alternative to the exponential, and is involved in the definition and representation of
fundamental physical concepts in thermodynamics, such as entropy and chemical
potential. The Arrhenius and Nernst equations, (1.3) and (1.4), provide an example of
the dual relation of the inverse functions. Another example is given in Figure 3.1(iv)
by alternative equations for the logarithmic spiral observed in natural phenomena, as
in the growth of shells of molluscs, flight behaviour of some birds and, on a larger
scale, the shape of hurricanes and some galaxies.
3.2 Trigonometric functions
Geometric definitions
The principal trigonometric functions of the angle θ, the internal angle at A of the right-
angle triangle in Figure 3.2, are the sine(sin), the cosine(cos), and the tangent (tan):
(3.1)
(3.2)
(3.3)
tansin
cos
θ
θ
θ
===
opposite
adjacent
BC
AB
cosθ==
adjacent
hypotenuse
AB
AC
sinθ==
opposite
hypotenuse
BC
AC
1The earliest ‘trigonometric tables’ were constructed in about 150 BC by the astronomer Hipparchus of Nicaea
(modern Iznik in Turkey), to whom we owe the 360° circle, and by Claudius Ptolemy of Alexandria (c.100–178
AD) whose Syntaxis mathematica(Mathematical Synthesis), known as the Almagest, was called by the Arabs
al-magisti(the greatest) and whose tables were used by astronomers for over a thousand years. The tables in
the Hindu Siddhantas(about 400 AD) are essentially tabulations of the sine function. Arab mathematicians
(about 950 AD) added new tabulations and theorems. European trigonometry was developed by Johann Müller
(Regiomontanus) of Königsberg (1436–1476), and by Georg Joachim Rheticus (1514–1576) of Wittenburg, a
student of Copernicus, whose Opus palatinum de triangulis(The palatine work on triangles), 1596, focused for
the first time on the properties of the right-angled triangle. François Viète (1540–1603) built on this work with
extensive new tables for all six common functions, new formulas, and the use of trigonometric functions for the
solution of algebraic equations. The word ‘trigonometry’ came into use in about 1600.
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hyp otenu se opposi te
adjacent
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Figure 3.2