Higher Engineering Mathematics, Sixth Edition

Trigonometric waveforms 139

y

1.0

2 1.0

(^0) A°
y 5 sin A y 5 sin 2A
90° 180° 270° 360°
Figure 14.12
(ii) Agraphofy=sin^12 AisshowninFig.14.13using
the following table of values.
A◦^12 A sin^12 A
0 0 0
30 15 0.259
60 30 0.500
90 45 0.707
120 60 0.866
150 75 0.966
180 90 1.00
210 105 0.966
240 120 0.866
270 135 0.707
300 150 0.500
330 165 0.259
360 180 0
y
1.0
2 1.0
0
1
2
A°
y 5 sin A y 5 sin A
90° 180° 270° 360°
Figure 14.13
y
(^0) A 8
2 1.0
1.0
y 5 cos A y^5 cos 2A
908 1808 2708 3608
Figure 14.14
(iii) A graph ofy=cosAis shown by the broken line
in Fig. 14.14 and is obtained by drawing up a
table of values. A similar table may be produced
fory=cos2Awith the result as shown.
(iv) A graph ofy=cos^12 Ais shown in Fig. 14.15
which may be produced by drawing up a table
of values, similar to above.
y
1.0
2 1.0
(^03608) A 8
1
2
y 5 cos A y^5 cos A
908 1808 2708
Figure 14.15
Periodic functionsand period
(i) Each of the graphs shown in Figs. 14.12 to 14.15
will repeat themselves as angle A increases and
are thus calledperiodic functions.
(ii) y=sinAandy=cosArepeat themselves every
360 ◦ (or 2πradians); thus 360◦is called the
period of these waveforms. y=sin2A and
y=cos2A repeat themselves every 180◦ (or
π radians); thus 180◦ is the period of these
waveforms.
(iii) In general, ify=sinpAory=cospA(wherep
is a constant) then the period of the waveform is
360 ◦/p(or 2π/prad). Hence ify=sin3Athen
the period is 360/3, i.e. 120◦,andify=cos4A
then the period is 360/4, i.e. 90◦.