Basic Engineering Mathematics, Fifth Edition

202 Basic Engineering Mathematics

Now try the following Practice Exercise

PracticeExercise 88 Trigonometric

waveforms (answers on page 350)

1. A sine wave is given byy=5sin3x. State its
   peak value.


2. A sine wave is given byy=4sin2x. State its
   period in degrees.


3. Aperiodicfunctionisgivenbyy=30cos5x.
   State its maximum value.


4. Aperiodicfunctionisgivenbyy=25cos3x.
   State its period in degrees.
   In problems 5 to 11, state the amplitude and period
   of the waveform and sketch the curve between 0◦
   and 360◦.


5. y=cos3A 6. y=2sin

5 x

2

7. y=3sin4t 8. y=5cos

θ

2

9. y=

7

2

sin

3 x

8

10. y=6sin(t− 45 ◦)


11. y=4cos( 2 θ+ 30 ◦)


12. The frequency of a sine wave is 200Hz.
    Calculate the periodic time.


13. Calculate the frequency of a sine wave that
    has a periodic time of 25ms.


14. Calculate the periodic time for a sine wave
    having a frequency of 10kHz.


15. An alternating current completes 15 cycles in
    24ms. Determine its frequency.


16. Graphs ofy 1 =2sinxand
    y 2 =3sin(x+ 50 ◦)are drawn on the same
    axes. Isy 2 lagging or leadingy 1?


17. Graphs ofy 1 =6sinxand
    y 2 =5sin(x− 70 ◦)are drawn on the same
    axes. Isy 1 lagging or leadingy 2?

22.5 Sinusoidal form:Asin(ωt±α)

Ifasinewaveisexpressedintheform

y=Asin(ωt±α)then

(a) A=amplitude.

(b) ω=angular velocity= 2 πfrad/s.

(c) frequency,f=

ω

2 π

hertz.

(d) periodic time,T=

2 π

ω

seconds

(

i.e.T=

1

f

)

.

(e) α= angle of lead or lag (compared with

y=Asinωt).

Here are some worked problems involving the sinu-

soidal formAsin(ωt±α).

Problem 10. An alternating current is given by

i=30sin( 100 πt+ 0. 35 )amperes. Find the

(a) amplitude, (b) frequency, (c) periodic time and

(d) phase angle (in degrees and minutes)

(a) i=30sin( 100 πt+ 0. 35 )A; hence,

amplitude=30A.

(b) Angular velocity,ω= 100 π,rad/s, hence

frequency,f=

ω

2 π

=

100 π

2 π

=50Hz

(c) Periodic time,T=

1

f

=

1

50

=0.02sor20ms.

(d) 0.35 is the angle inradians. The relationship

between radians and degrees is

360 ◦= 2 πradians or 180 ◦=πradians

from which,

1 ◦=

π

180

radand1rad=

180 ◦

π

(≈ 57. 30 ◦)

Hence,phase angle,α= 0 .35 rad

=

(

0. 35 ×

180

π

)◦

=20.05◦or 20 ◦ 3 ′leading

i=30sin(100πt).

Problem 11. An oscillating mechanism has a

maximum displacement of 2.5m and a frequency of

60Hz. At timet=0 the displacement is 90cm.

Express the displacement in the general form

Asin(ωt±α)

Amplitude=maximum displacement=2.5m.

Angular velocity,ω= 2 πf= 2 π( 60 )= 120 πrad/s.

Hence, displacement= 2 .5sin( 120 πt+α)m.

Whent=0, displacement=90cm= 0 .90m.