Higher Engineering Mathematics, Sixth Edition

166 Higher Engineering Mathematics

There is only one quadrant where both sinαandcosα

are positive, and this is the first, as shown in Fig. 17.2.

From Fig. 17.2, by Pythagoras’ theorem:

R=

√

( 32 + 42 )= 5

R 4

3



Figure 17.2

From trigonometric ratios: α=tan−^143 = 53. 13 ◦ or

0.927 radians.

Hence 3sinωt+4cosωt=5sin(ω t+ 0. 927 ).

Asketchof3sinωt,4cosωtand 5sin(ωt+ 0. 927 )is

shown in Fig. 17.3.

Two periodic functions of the same frequency may be

combined by,

(a) plotting the functions graphically and combining

ordinates at intervals, or

(b) byresolutionofphasorsbydrawingorcalculation.

Problem 6, together with Problems 7 and 8 following,

demonstrate a third method of combining waveforms.

Problem 7. Express 4.6sinωt− 7 .3cosωtin the

formRsin(ωt+α).

Let 4.6sinωt− 7 .3cosωt=Rsin(ωt+α).

then 4.6sinωt− 7 .3cosωt

=R[sinωtcosα+cosωtsinα]

=(Rcosα)sinωt+(Rsinα)cosωt

Equating coefficients of sinωtgives:

4. 6 =Rcosα,from which, cosα=

4. 6

R

Equating coefficients of cosωtgives:

− 7. 3 =Rsinα,from which,sinα=

− 7. 3

R

There is only one quadrant where cosine is positiveand

sine is negative, i.e. the fourth quadrant, as shown in

Fig. 17.4. By Pythagoras’ theorem:

R=

√

[( 4. 6 )^2 +(− 7. 3 )^2 ]= 8. 628

By trigonometric ratios:

α=tan−^1

(

− 7. 3

4. 6

)

=− 57. 78 ◦or− 1 .008 radians.

Hence

4 .6sinωt− 7 .3cosωt= 8 .628sin(ω t− 1. 008 ).

y 5 4cost

y 5 3sint

y 5 5 sin(t 1 0.927)

0.927 rad

0.927 rad

0  t (rad)

23

21

22

24

25

3

1

2

4

5

y

/2 3/2 2 

Figure 17.3