Higher Engineering Mathematics

TRIGONOMETRIC WAVEFORMS 157

B

Figure 15.28

Now try the following exercise.

Exercise 70 Further problems on sine and

cosine curves

In Problems 1 to 9 state the amplitude and period

of the waveform and sketch the curve between

0 ◦and 360◦.

1.y=cos 3A [1, 120◦]

2.y=2 sin

5 x

2

[2, 144◦]

3.y=3 sin 4t [3, 90◦]

4.y=3 cos

θ

2

[3, 720◦]

5.y=

7

2

sin

3 x

8

[

7

2

, 960◦

]

6.y=6 sin(t− 45 ◦) [6, 360◦]

7.y=4 cos(2θ+ 30 ◦) [4, 180◦]

8.y=2 sin^22 t [2, 90◦]

9.y=5 cos^2

3

2

θ [5, 120◦]

Figure 15.29

15.5 Sinusoidal formAsin(ωt±α)

In Figure 15.29, letORrepresent a vector that is

free to rotate anticlockwise aboutOat a velocity of

ωrad/s. A rotating vector is called aphasor. After

a timetsecondsORwill have turned through an

angleωtradians (shown as angleTORin Fig. 15.29).

If ST is constructed perpendicular to OR, then

sinωt=ST/TO, i.e.ST=TOsinωt.

If all such vertical components are projected on

to a graph ofyagainstωt, a sine wave results of

amplitudeOR(as shown in Section 15.3).

If phasor OR makes one revolution (i.e. 2π

radians) inTseconds, then the angular velocity,

ω= 2 π/Trad/s, from which, T= 2 π/ωseconds.

Tis known as theperiodic time.

The number of complete cycles occurring per

second is called thefrequency,f

Frequency=

number of cycles

second

=

1

T

=

ω

2 π

i.e. f=

ω

2 π

Hz

Henceangular velocity, ω= 2 πfrad/s

Amplitude is the name given to the maximum

or peak value of a sine wave, as explained in

Section 15.4. The amplitude of the sine wave shown

in Fig. 15.29 has an amplitude of 1.

A sine or cosine wave may not always start at

0 ◦. To show this a periodic function is represented

byy=sin (ωt±α)ory=cos (ωt±α), whereαis

a phase displacement compared withy=sinAor

y=cosA. A graph ofy=sin (ωt−α)lagsy=sinωt