Higher Engineering Mathematics, Sixth Edition

Trigonometric waveforms 141

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

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

Fig. 14.20. 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◦.

908 2708

y

(^0) A 8
2 1.0
1.0 y^5 sin(A^2608 )
y 5 sin A
608
608
1808 3608
Figure 14.20
(iii) By drawing up a table of values, a graph of
y=cos(A+ 45 ◦)may be plotted as shown in
Fig. 14.21. 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◦.
1808
458
(^03608) A 8
y
2 1.0
y 5 cos (A 1458 )
y 5 cos A
458
908 2708
Figure 14.21
(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. Sketchy=5sin(A+ 30 ◦)from
A= 0 ◦toA= 360 ◦.
Amplitude=5; period= 360 ◦/ 1 = 360 ◦.
5sin(A+ 30 ◦)leads 5sinAby 30◦(i.e. starts 30◦
earlier).
Asketchofy=5sin(A+ 30 ◦)is shown in Fig. 14.22.
(^09082708) A 8
5
25
y 5 5 sin(A 1308 )
y 5 5 sin A
308
308
(^18083608)
y
Figure 14.22
Problem 10. Sketchy=7sin( 2 A−π/ 3 )in the
range 0≤A≤ 2 π.
Amplitude=7; period= 2 π/ 2 =πradians.
In general, y=sin(pt−α) lags y=sinpt by α/p,
hence 7 sin( 2 A−π/ 3 ) lags 7sin2A by (π/ 3 )/2,
i.e.π/6rad or 30◦.
Asketchofy=7sin( 2 A−π/ 3 )is shown in Fig. 14.23.
0
7
y
(^3608) A 8
y 5 7sin 2A
y 5 7 sin(2A 2 /3)
/6
/6
 2 
7
908 1808 2708
/ 2 3 / 2
Figure 14.23
Problem 11. Sketchy=2cos(ωt− 3 π/ 10 )over
one cycle.
Amplitude=2; period= 2 π/ωrad.
2cos(ωt− 3 π/ 10 )lags 2cosωtby 3π/ 10 ωseconds.
Asketchofy=2cos(ωt− 3 π/ 10 ) is shown in
Fig. 14.24.