200 Basic Engineering Mathematics

y

1.0

1.0

0 90  180  270  360 A

ysin 3A

Figure 22.14

y

0 90  180  270  360  x

 4

4 y4 cos 2x

Figure 22.15

Problem 4. Sketchy=2sin

3

5

Aover one cycle

Amplitude= 2 ;period=

360 ◦

3

5

=

360 ◦× 5

3

= 600 ◦

Asketchofy=2sin

3

5

Ais shown in Figure 22.16.

180  360  540  600 

y

(^0) A
 2
2
y2 sin^35 A
Figure 22.16
22.4.4 Periodic time
In practice, the horizontal axis of a sine wave will be
time. The time taken for a sine wave to complete one
cycle is called theperiodic time,T.
In the sine wave of voltagev(volts) against timet(mil-
liseconds) shown in Figure 22.17, the amplitude is 10V
and the periodic time is 20ms; i.e.,T=20ms.
v
(^01020) t (ms)
210
10
Figure 22.17
22.4.5 Frequency
The number of cycles completed in one second is called
thefrequencyfand is measured inhertz,Hz.
f=
1
T
orT=
1
f
Problem 5. Determine the frequency of the sine
wave shown in Figure 22.17
In the sine wave shown in Figure 22.17,T=20ms,
hence
frequency,f=
1
T

1
20 × 10 −^3
=50Hz
Problem 6. If a waveform has a frequency of
200kHz, determine the periodic time
If a waveform has a frequency of 200kHz, the periodic
timeTis given by
periodic time,T=
1
f

1
200 × 103
= 5 × 10 −^6 s= 5 μs
22.4.6 Lagging and leading angles
A sine or cosine curve may not always start at 0◦.
To show this, a periodic function is represented by
y=Asin(x±α)where α is a phase displacement
compared withy=Asinx. For example,y=sinAis
shown by the broken line in Figure 22.18 and, on
thesameaxes,y=sin(A− 60 ◦)is shown.The graph
y=sin(A− 60 ◦)is said to lagy=sinAby 60◦.
In another example,y=cosAis shown by the bro-
ken line in Figure 22.19 and, on the same axes,y=
cos(A+ 45 ◦)is shown.The graphy=cos(A+ 45 ◦)is
said to leady=cosAby 45◦.