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 ¢¡£¡¥¤§¦¨¤§©  £
 £© Trigonometric Functions
Many processes in nature are cyclic. A pendulum oscillates back and forth, repeating its
motion over and over; a weight hanging at the end of a spring bobs up and down; the Earth
repeats its orbit about the Sun every 365 days; a population of arctic wolves has periods
of growth followed by periods of decrease, following the fluctuations in the population of
their prey; the monthly rainfall at an agricultural research station varies cyclically over
the years and over the decades. To model such natural behavior, a mathematician needs
functions which repeat their values over intervals of fixed length. These functions are the
periodic functions. Precisely, a function f is periodic if there is a fixed constant T such
that f(t + T ) = f(t) for every value of t in the domain of f. The smallest such positive T
for which this property holds is called the period of f.
c b
θ
a
Figure A right triangle
The class of periodic functions that we will consider in this section are the trigonometric
functions. Although these functions were originally invented to work with problems of
measurement, their importance in modern mathematics stems more from their periodic
behavior. We will begin with a definition in terms of measuring the sides of a right
triangle. Consider a right triangle with legs of lengths a and b and hypotenuse of length c.
Moreover, suppose, as in Figure , the angle opposite the leg of length b has measure
θ. Then we define the sine of θ, which we write as sin(θ), by
b
sin(θ) = ()
c
and the cosine of θ, which we write as cos(θ), by
a
cos(θ) = . ()
c
1
2 Trigonometric Functions Section
(0, 1)
(cos(θ), sin(θ))
θ
b
(-1, 0) a (1, 0)
(0, -1)
Figure A right triangle with a vertex on the unit circle
The properties of similar triangles, known even by the ancient Egyptians and