Chapter 10 Trigonometric functions - Ugrad.math.ubc.ca

Math 102 Notes Chapter 10
1
y=sin (t)
0
−1
π/2 π 3π/2 2π 5π/2
3π
t
period, T
y=cos (t)
1
0
π/2 π 3π/2 2π 5π/2 3π
t
−1
period, T
Figure 10.5: Periodicity of the sine and cosine. Note that the two curves are just shifted versions
of one another.
10.4 Phase, amplitude, and frequency
We have already learned how the appearance of functions changes when we shift their graph in
one direction or another, scale one of the axes, and so on. Thus it will be easy to follow the basic
changes in shape of a typical trigonometric function.
A function of the form
y = f(t) = A sin(ωt)
has both its t and y axes scaled. The constant A, referred to as the amplitude of the graph, scales
the y axis so that the oscillation swings between a low value of −A and a high value of A. The
constant ω, called the frequency, scales the t axis. This results in crowding together of the peaks
and valleys (if ω > 1) or stretching them out (if ω < 1). One full cycle is completed when
ωt = 2π
and this occurs at time
t = 2π ω .
We will use the symbol T, to denote this special time, and we refer to T as the period. We note the
connection between frequency and period:
ω = 2π
T ,
T = 2π ω .
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