140 Higher Engineering Mathematics

Amplitude

Amplitude is the name given to the maximum or peak

value of a sine wave. Each of the graphs shown in

Figs. 14.12 to 14.15 has an amplitude of+1 (i.e. they

oscillate between+1and−1). However, ify=4sinA,

each of the values in the table is multiplied by 4 and

the maximum value, and thus amplitude, is 4. Simi-

larly, ify=5cos2A, theamplitude is 5 and the period is

360 ◦/2,i.e.180◦.

Problem 5. Sketchy=sin3AbetweenA= 0 ◦

andA= 360 ◦.

Amplitude= 1 ;period= 360 ◦/ 3 = 120 ◦.

Asketchofy=sin3Ais shown in Fig. 14.16.

y

1.0

2 1.0

(^0908180827083608) A 8
y 5 sin 3A
Figure 14.16
Problem 6. Sketchy=3sin2AfromA=0to
A= 2 πradians.
Amplitude= 3 ,period= 2 π/ 2 =πrads (or 180◦).
Asketchofy=3sin2Ais shown in Fig. 14.17.
y
3
23
(^0) A 8
y 5 3 sin 2A
908 1808 2708 3608
Figure 14.17
Problem 7. Sketchy=4cos2xfromx= 0 ◦to
x= 360 ◦.
Amplitude= 4 ;period= 360 ◦/ 2 = 180 ◦.
Asketchofy=4cos2xis shown in Fig. 14.18.
y
(^0908180827083608) x 8
24
4 y^5 4 cos 2x
Figure 14.18
Problem 8. Sketchy=2sin
3
5
Aover one cycle.
Amplitude= 2 ;period=
360 ◦
3
5

360 ◦× 5
3
= 600 ◦.
Asketchofy=2sin
3
5
Ais shown in Fig. 14.19.
1808 3608 5408 6008
y
(^0) A 8
22
(^2) y 5 2 sin (^3) A
5
Figure 14.19
Lagging and leading angles
(i) A sine or cosine curve may not always start at 0◦.
To show this a periodic function is represented
by y=sin(A±α)or y=cos(A±α)where α
is a phase displacement compared withy=sinA
ory=cosA.