Basic Engineering Mathematics, Fifth Edition

Trigonometric waveforms 199

2 0.5

0.5

2 1.0

1.0

y

S

R

T S^9

O 9

y 5 cosx

Angle x 8

308 608 1208 1808 2408 3008 3608

458

15808

3308

3158

2858

2558

08

2258

2108

1808

1508

1208

908

608

O

Figure 22.12

22.3.1 Sine waves

The vertical componentTSmay be projected across to
T′S′, which is the corresponding value of 30◦on the
graph ofyagainst anglex◦. If all such vertical compo-
nents asTSare projected on to the graph, asine wave
is produced as shown in Figure 22.11.

22.3.2 Cosine waves

If all horizontal components such asOSare projected
on to a graph ofyagainst anglex◦,acosine waveis
produced. It is easier to visualize these projections by
redrawing the circle with the radius armORinitially in
a vertical position as shown in Figure 22.12.
It is seen from Figures 22.11 and 22.12 that a cosine
curve is of the same form as the sine curve but is dis-
placed by 90◦(orπ/2 radians). Both sine and cosine
waves repeat every 360◦.

22.4 Terminology involved with sine

and cosine waves

Sine waves are extremely important in engineering, with
examples occurring with alternating currents and volt-
ages – the mains supply is a sine wave – and with simple
harmonic motion.

22.4.1 Cycle

When a sine wave has passed through a complete
series of values, both positive and negative, it is said
to have completed onecycle. One cycle of a sine
wave is shown in Figure 22.1(a) on page 195 and in
Figure 22.11.

22.4.2 Amplitude

The amplitude is the maximum value reached in a half

cycle by a sine wave. Another name foramplitudeis

peak valueormaximum value.

Asinewavey=5sinx has an amplitude of 5, a

sine wavev=200sin314thas an amplitude of 200 and

the sine wavey=sinxshown in Figure 22.11 has an

amplitude of 1.

22.4.3 Period

The waveformsy=sinxandy=cosxrepeat them-

selves every 360◦. Thus, for each, theperiodis 360◦.

Awaveformofy=tanxhas a period of 180◦(from

Figure 22.1(c)).

A graph ofy=3sin2A, as shown in Figure 22.13, has

anamplitude of 3andperiod 180◦.

A graph ofy=sin3A, as shown in Figure 22.14, has an

amplitude of 1andperiod of 120◦.

A graph ofy=4cos2x, as shown in Figure 22.15, has

an amplitude of 4 and a period of 180◦.

In general, if y=Asinpx or y=Acospx,

amplitude=Aand period=

360 ◦

p

y

3

23

(^0) A 8
y 5 3 sin 2A
908 1808 2708 3608
Figure 22.13