Higher Engineering Mathematics

TRIGONOMETRIC WAVEFORMS 159

B

(c)Whent=10 ms

thenv=340 sin

(

50 π

10

103

− 0. 541

)

=340 sin(1.0298)=340 sin 59◦

=291.4 V

(d) Whenv=200 volts

then 200=340 sin(50πt− 0 .541)

200

340

=sin(50πt− 0 .541)

Hence (50πt− 0 .541)=arcsin

200

340

= 36. 03 ◦or 0.6288 rad

50 πt= 0. 6288 + 0. 541

= 1. 1698

Hence whenv=200 V,

time,t=

1. 1698

50 π

=7.447 ms

(e) When the voltage is a maximum,v=340 V.

Hence 340 =340 sin (50πt− 0 .541)

1 =sin (50πt− 0 .541)

50 πt− 0. 541 =arcsin 1

= 90 ◦or 1.5708 rad

50 πt= 1. 5708 + 0. 541 = 2. 1118

Hence time,t=

2. 1118

50 π

=13.44 ms

A sketch ofv=340 sin(50πt− 0 .541) volts is shown

in Fig. 15.30.

Figure 15.30

Now try the following exercise.

Exercise 71 Further problems on the

sinusoidal formAsin(ωt±α)

In Problems 1 to 3 find the amplitude, peri-

odic time, frequency and phase angle (stating

whether it is leading or laggingAsinωt)ofthe

alternating quantities given.

1.i=40 sin (50πt+ 0 .29) mA

[

40, 0.04 s, 25 Hz, 0.29 rad

(or 16◦ 37 ′) leading 40 sin 50πt

]

2.y=75 sin (40t− 0 .54) cm

[

75 cm, 0.157 s, 6.37 Hz, 0.54 rad

(or 30◦ 56 ′) lagging 75 sin 40t

]

3.v=300 sin (200[ πt− 0 .412) V

300 V, 0.01 s, 100 Hz, 0.412 rad

(or 23◦ 36 ′) lagging 300 sin 200πt

]

4. A sinusoidal voltage has a maximum value
   of 120 V and a frequency of 50 Hz. At time
   t=0, the voltage is (a) zero, and (b) 50 V.
   Express the instantaneous voltagevin the
   formv=Asin(ωt±α)
   [
   (a)v=120 sin 100πtvolts
   (b)v=120sin(100πt+ 0 .43) volts

]

5. An alternating current has a periodic time of
   25 ms and a maximum value of 20 A. When
   timet=0, currenti=−10 amperes. Express
   the currentiin the formi=Asin(ωt±α)
   [
   i=20 sin

(

80 πt−

π

6

)

amperes

]

6. An oscillating mechanism has a maximum
   displacement of 3.2 m and a frequency of
   50 Hz. At time t=0 the displacement is
   150 cm. Express the displacement in the gen-
   eral formAsin(ωt±α).

[3.2 sin(100πt+ 0 .488) m]

7. The current in an a.c. circuit at any time
   tseconds is given by:
   i=5 sin(100πt− 0 .432) amperes
   Determine (a) the amplitude, periodic time,
   frequency and phase angle (in degrees) (b) the
   value of current att=0 (c) the value of