Higher Engineering Mathematics

TRIGONOMETRIC WAVEFORMS 153

B

Figure 15.12

A graph ofy=sin 2Ais shown in Fig. 15.12.

(ii) A graph ofy=sin^12 Ais shown in Fig. 15.13

using the following table of values.

A◦^12 A sin^12 A

000

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

Figure 15.13

(iii) A graph ofy=cosAis shown by the broken line
in Fig. 15.14 and is obtained by drawing up a

Figure 15.14

table of values. A similar table may be produced

fory=cos 2Awith the result as shown.

(iv) A graph ofy=cos^12 Ais shown in Fig. 15.15

which may be produced by drawing up a table

of values, similar to above.

Figure 15.15

Periodic functions and period

(i) Each of the graphs shown in Figs. 15.12

to 15.15 will repeat themselves as angle

A increases and are thus called periodic

functions.

(ii)y=sinAandy=cosArepeat themselves every

360 ◦(or 2πradians); thus 360◦is called the

period of these waveforms. y=sin 2A and

y=cos 2Arepeat 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=sin 3Athen

the period is 360/3, i.e. 120◦, and ify=cos 4A

then the period is 360/4, i.e. 90◦.