Higher Engineering Mathematics, Sixth Edition

266 Higher Engineering Mathematics

amplitude, of the resultant is 3.6. The resultant wave-

formleadsy 1 =3sinAby34◦or34×

π

180

rad= 0. 593

rad.

The sinusoidal expression for the resultant wave-

form is:

yR= 3 .6sin(A+ 34 ◦) or

yR= 3 .6sin(A+ 0. 593 )

Problem 2. Plot the graphs ofy 1 =4sinωtand

y 2 =3sin(ωt−π/ 3 )on the same axes, over one

cycle. By adding ordinates at intervals plot

yR=y 1 +y 2 and obtain a sinusoidal expression for

the resultant waveform.

y 1 =4sinωtandy 2 =3sin(ωt−π/ 3 )are shown plot-

ted in Fig. 25.2.

908

y

y 15 4 sint

y 25 3 sin (t 2 /3)

0

26

24

22

6

6.1

4

2

258

258

yR 5 y 11 y 2

(^180827083608) t
/2  3 /2 2 
Figure 25.2
Ordinates are added at 15◦intervals and the resul-
tant is shown by the broken line. The amplitude
of the resultant is 6.1 and it lags y 1 by 25◦
or 0.436 rad.
Hence, the sinusoidal expression for the resultant wave-
form is:
yR= 6 .1sin(ωt− 0. 436 )
Problem 3. Determine a sinusoidal expression
fory 1 −y 2 wheny 1 =4sinωtand
y 2 =3sin(ωt−π/ 3 ).
y 1 andy 2 are shown plottedinFig. 25.3. At 15◦intervals
y 2 is subtracted fromy 1. For example:
at 0◦,y 1 −y 2 = 0 −(− 2. 6 )=+ 2. 6
at 30◦,y 1 −y 2 = 2 −(− 1. 5 )=+ 3. 5
at 150◦,y 1 −y 2 = 2 − 3 =− 1 ,and so on.
908
y
y 1
0
24
22
4
2
3.6
458
y 2
y 12 y 2
(^180827083608) t
/2  3 /2 2 
Figure 25.3
The amplitude, or peak value of the resultant (shown by
the broken line), is 3.6 and it leadsy 1 by 45◦or 0.79
rad. Hence,
y 1 −y 2 = 3 .6sin(ωt+ 0. 79 )
Problem 4. Two alternating currents are given by:
i 1 =20sinωtamperes and
i 2 =10sin
(
ωt+
π
3
)
amperes.
By drawing the waveforms on the same axes and
adding, determine the sinusoidal expression for the
resultanti 1 +i 2.
i 1 andi 2 are shown plotted in Fig. 25.4. The resultant
waveform fori 1 +i 2 is shown by the broken line. It has
the same period, and hence frequency, asi 1 andi 2.
2  angle t
198
198
i 15 20 sint
908 1808 2708 3608
230
220
210
10
20
26.5
30
3 
2

2

iR 5 20 sint 1 10 sin (t 1 3 )
i 25 10 sin (t 1 3 )
Figure 25.4
The amplitude or peak value is 26.5 A.
The resultant waveform leads the waveform of
i 1 =20sinωtby 19◦or 0.33 rad