Higher Engineering Mathematics

154 GEOMETRY AND TRIGONOMETRY

Amplitude

Amplitude is the name given to the maximum or
peak value of a sine wave. Each of the graphs
shown in Figs. 15.12 to 15.15 has an amplitude of
+1 (i.e. they oscillate between+1 and−1). How-
ever, ify=4 sinA, each of the values in the table
is multiplied by 4 and the maximum value, and
thus amplitude, is 4. Similarly, ify=5 cos 2A, the
amplitude is 5 and the period is 360◦/2, i.e. 180◦.

Problem 5. Sketchy=sin 3AbetweenA= 0 ◦

andA= 360 ◦.

Amplitude=1; period= 360 ◦/ 3 = 120 ◦.

A sketch ofy=sin 3Ais shown in Fig. 15.16.

Figure 15.16

Problem 6. Sketchy=3 sin 2AfromA=0to

A= 2 πradians.

Amplitude=3, period= 2 π/ 2 =πrads (or 180◦).

A sketch ofy=3 sin 2Ais shown in Fig. 15.17.

Figure 15.17

Problem 7. Sketchy=4 cos 2xfromx= 0 ◦to

x= 360 ◦.

Amplitude=4; period= 360 ◦/ 2 = 180 ◦.

A sketch ofy=4 cos 2xis shown in Fig. 15.18.

Figure 15.18

Problem 8. Sketchy=2 sin

3

5

Aover one

cycle.

Amplitude=2; period=

360 ◦

3

5

=

360 ◦× 5

3

= 600 ◦.

A sketch ofy=2 sin

3

5

Ais shown in Fig. 15.19.

Figure 15.19

Lagging and leading angles

(i) A sine or cosine curve may not always start

at 0◦. To show this a periodic function is rep-

resented byy=sin(A±α)ory=cos(A±α)