Higher Engineering Mathematics

TRIGONOMETRIC WAVEFORMS 155

B

whereαis a phase displacement compared with

y=sinAory=cosA.

(ii) By drawing up a table of values, a graph of

y=sin(A− 60 ◦) may be plotted as shown in

Fig. 15.20. Ify=sinAis assumed to start at 0◦

theny=sin(A− 60 ◦) starts 60◦later (i.e. has a

zero value 60◦later). Thusy=sin(A− 60 ◦)is

said tolagy=sinAby 60◦.

Figure 15.20

(iii) By drawing up a table of values, a graph of
y=cos(A+ 45 ◦) may be plotted as shown in
Fig. 15.21. Ify=cosAis assumed to start at 0◦
theny=cos(A+ 45 ◦) starts 45◦earlier (i.e. has
a zero value 45◦earlier). Thusy=cos(A+ 45 ◦)
is said toleady=cosAby 45◦.

Figure 15.21

(iv) Generally, a graph ofy=sin(A−α) lags

y=sinAby angleα, and a graph of

y=sin(A+α) leadsy=sinAby angleα.

(v) A cosine curve is the same shape as a sine curve

but starts 90◦earlier, i.e. leads by 90◦. Hence

cosA=sin(A+ 90 ◦).

Problem 9. Sketch y=5 sin(A+ 30 ◦) from

A= 0 ◦toA= 360 ◦.

Amplitude=5; period= 360 ◦/ 1 = 360 ◦.

5 sin(A+ 30 ◦) leads 5 sinAby 30◦(i.e. starts 30◦

earlier).

A sketch ofy=5 sin(A+ 30 ◦) is shown in Fig. 15.22.

Figure 15.22

Problem 10. Sketchy=7 sin(2A−π/3) in the

range 0≤A≤ 2 π.

Amplitude=7; period= 2 π/ 2 =πradians.

In general, y=sin(pt−α) lagsy=sinptbyα/p,

hence 7 sin(2A−π/3) lags 7 sin 2A by (π/3)/2,

i.e.π/6 rad or 30◦.

A sketch of y=7 sin(2A−π/3) is shown in

Fig. 15.23.

Figure 15.23