082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

132 Chapter 3 The trigonometric functions
to 2π seconds, the angle ωt increases from 0 to 2π radians. We know that as the angle
ω
ωt increases by 2π radians then Asin ωt completes a full cycle. Hence a full cycle is
completed in 2π ω seconds, thatisthe period ofy =Asin ωt is 2π ω seconds.
Ify =Asin ωt ory =Acos ωt, thenthe periodT is 2π ω .
Inparticular wenote thatthe period ofy =Asint andy =Acost is2π.
Closelyrelatedtotheperiodisthefrequencyofawave.Thefrequencyisthenumber
ofcyclescompletedin1second.Frequency ismeasuredinunitscalledhertz(Hz).One
hertz isone cycle per second. We have seen thaty =Asin ωt takes
2π
seconds tocomplete one cycle
ω
and soitwill take
1 second tocomplete ω 2π cycles
We use f as the symbol forfrequency and so
frequency, f = ω 2π
( ) 3
For example, sin3t has a frequency of Hz.
2π
Note that by rearrangement wemay write
ω=2πf
and sothe wavey =Asin ωt may alsobewritten asy =Asin2πft.
Fromthe definitions of period and frequency wecan see that
1
period =
frequency
that is
T = 1 f
We see that the period is the reciprocal of the frequency. Identical results apply for the
wavey =Acosωt.
A final generalization is to introduce a phase angle or phase, φ. This allows the
wave to be shifted along the time axis. It also means that either a sine function or a
cosine function can be usedtorepresent the same wave. So the general formsare
Acos(ωt + φ), Asin(ωt + φ)
Figure 3.16 depictsAsin(ωt + φ). Note from Figure 3.16 that the actual movement of
the wave along the time axis is φ/ω. Itis easytoshow thismathematically:
(
Asin(ωt +φ) =Asinω t + φ )
ω
The quantity φ iscalled thetimedisplacement.
ω