Higher Engineering Mathematics, Sixth Edition

Trigonometric waveforms 145

(c) Whent=10ms

thenv=340sin

(

50 π

10

103

− 0. 541

)

=340sin( 1. 0298 )=340sin59◦

=291.4V

(d) Whenv=200volts

then 200=340sin( 50 πt− 0. 541 )

200

340

=sin( 50 πt− 0. 541 )

Hence( 50 πt− 0. 541 )=arcsin

200

340

= 36. 03 ◦or 0.6288rad

50 πt= 0. 6288 + 0. 541

= 1. 1698

Hence whenv=200V,

time,t=

1. 1698

50 π

=7.447ms

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

Hence 340=340sin( 50 πt− 0. 541 )

1 =sin( 50 πt− 0. 541 )

50 πt− 0. 541 =arcsin1

= 90 ◦or 1.5708rad

50 πt= 1. 5708 + 0. 541 = 2. 1118

Hence time,t=

2. 1118

50 π

=13.44ms

Asketchofv=340 sin( 50 πt− 0. 541 )volts is shown in
Fig. 14.30.

v 5 340 sin(50t 2 0.541)

v 5 340 sin 50t

(^0103040) t (ms)
7.447 13.44
2340
2 175.1
200
291.4
340
Voltage V
20
Figure 14.30
Now try the following exercise
Exercise 63 Further problemson the
sinusoidal formAsin(ωt±α)
In Problems 1 to 3 find the amplitude, periodic
time, frequency and phase angle (stating whether
it is leading or laggingAsinωt) of the alternating
quantities given.

1. i=40sin( 50 πt+ 0. 29 )mA
   [
   40 , 0 .04s,25Hz, 0 .29rad
   (or 16◦ 37 ′)leading 40sin50πt

]

2. y=75sin( 40 t− 0. 54 )cm
   [
   75cm, 0 .157s, 6 .37Hz, 0 .54rad
   (or30◦ 56 ′)lagging75sin 40t

]

3. v=300sin( 200 πt− 0. 412 )V
   [
   300V, 0 .01s,100Hz, 0 .412rad
   (or 23◦ 36 ′)lagging 300sin200πt

]

4. A sinusoidal voltage has a maximum value of
   120V and a frequency of 50Hz. At timet=0,
   the voltage is (a) zero, and (b) 50V.
   Express the instantaneous voltagevin the
   formv=Asin(ωt±α).
   [
   (a)v=120sin100πtvolts
   (b)v=120sin( 100 πt+ 0. 43 )volts

]

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

(

80 πt−

π

6

)

amperes

]

6. Anoscillatingmechanismhasamaximumdis-
   placement of 3.2m and a frequency of 50Hz.
   At timet=0 the displacement is 150cm.
   Express the displacement in the general form
   Asin(ωt±α).
   [3.2 sin( 100 πt+ 0. 488 )m]


7. The current in an a.c. circuit at any time
   tseconds is given by:

i=5sin( 100 πt− 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 current

att=8ms (d) the time when the current is first

a maximum (e) the time when the current first