COMPOUND ANGLES 179B
Figure 18.3
Hence 3 sinωt+4 cosωt=5 sin(ωt+ 0. 927 ).
A sketch of 3 sinωt, 4 cosωtand 5 sin(ωt+ 0 .927)
is shown in Fig. 18.3.
Two periodic functions of the same frequency may
be combined by,
(a) plotting the functions graphically and combin-
ing ordinates at intervals, or
(b) by resolution of phasors by drawing or
calculation.
Problem 6, together with Problems 7 and 8 fol-
lowing, demonstrate a third method of combining
waveforms.
Problem 7. Express 4.6 sinωt− 7 .3 cosωtin
the formRsin(ωt+α).Let 4.6 sinωt− 7 .3 cosωt=Rsin(ωt+α).
then 4.6 sinωt− 7 .3 cosωt
=R[sinωtcosα+cosωtsinα]
=(Rcosα) sinωt+(Rsinα) cosωtEquating coefficients of sinωtgives:
4. 6 =Rcosα, from which, cosα=4. 6
REquating coefficients of cosωtgives:− 7. 3 =Rsinα, from which, sinα=− 7. 3
RThere is only one quadrant where cosine is posi-
tiveandsine is negative, i.e., the fourth quadrant, as
shown in Fig. 18.4. By Pythagoras’ theorem:R=√
[(4.6)^2 +(− 7 .3)^2 ]= 8. 628Figure 18.4By trigonometric ratios:α=tan−^1(
− 7. 3
4. 6)=− 57. 78 ◦or− 1 .008 radians.