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

136 Chapter 3 The trigonometric functions
Engineering application 3.3 illustrates an important property when combining together
two sinusoidal waves of the same angular frequency.
If two waves of equal angular frequency, ω, areadded the resultisawave of the
same angular frequency, ω.
Infactthisresultholdstruewhencombininganynumberofwavesofthesameangular
frequency.
Engineeringapplication3.4
Combiningtwosinusoidalcurrentsignals
Two current signals,i 1
(t) andi 2
(t), have the following mathematical expressions:
i 1
(t) =10sin4t
i 2
(t) =5cos4t
(a) State the amplitude and angular frequency ofthe two signals.
(b) Obtain an expression forthe signal,i 3
(t), given byi 3
(t) = 0.3i 1
(t) −0.4i 2
(t).
(c) Reduce the expression obtained in part (b) to a single sinusoid in the form
Rcos(4t + φ) and hence stateits amplitude and phase.
Solution
(a) i 1
(t)hasanamplitudeof10ampsandanangularfrequency ω = 4 rad s −1 .i 2
(t)
has an amplitude of 5 amps and an angular frequency ω = 4 rad s −1 . Note that
both signals have the same angular frequency.
(b) i 3
(t) =0.3i 1
(t) −0.4i 2
(t)
(c) Let
=0.3×10sin4t−0.4×5cos4t
=3sin4t−2cos4t
3sin4t −2cos4t =Rcos(4t +φ)
Then using the trigonometric identity given inTable 3.1
cos(A +B) =cosAcosB −sinAsinB
withA=4tandB=φwefind
3sin4t −2cos4t =Rcos(4t +φ)
Hence
=R(cos4tcosφ −sin4tsinφ)
= (Rcosφ)cos4t −(Rsinφ)sin4t
3=−Rsinφ (3.3)
−2=Rcosφ (3.4)
Bysquaring each equation and adding we obtain
9+4=R 2 (sin 2 φ+cos 2 φ)=R 2
so thatR = √ 13.