Higher Engineering Mathematics

156 GEOMETRY AND TRIGONOMETRY

Problem 11. Sketch y=2 cos(ωt− 3 π/10)

over one cycle.

Amplitude=2; period= 2 π/ωrad.

2 cos(ωt− 3 π/10) lags 2 cosωtby 3π/ 10 ωseconds.

A sketch of y=2 cos(ωt− 3 π/10) is shown in
Fig. 15.24.

Figure 15.24

Graphs of sin^2 Aand cos^2 A

(i) A graph ofy=sin^2 Ais shown in Fig. 15.25

using the following table of values.

A◦ sinA (sin A)^2 =sin^2 A

00 0

30 0.50 0.25

60 0.866 0.75

90 1.0 1.0

120 0.866 0.75

150 0.50 0.25

180 0 0

210 −0.50 0.25

240 −0.866 0.75

270 −1.0 1.0

300 −0.866 0.75

330 −0.50 0.25

360 0 0

(ii) A graph ofy=cos^2 Ais shown in Fig. 15.26

obtained by drawing up a table of values, similar

to above.

(iii)y=sin^2 A and y=cos^2 A are both periodic
functions of period 180◦(orπrad) and both

Figure 15.25

Figure 15.26

contain only positive values. Thus a graph of

y=sin^22 Ahas a period 180◦/2, i.e., 90◦. Simi-

larly, a graph ofy=4 cos^23 Ahas a maximum

value of 4 and a period of 180◦/3, i.e. 60◦.

Problem 12. Sketchy=3 sin^212 Ain the range

0 <A< 360 ◦.

Maximum value=3; period= 180 ◦/(1/2)= 360 ◦.

A sketch of 3 sin^212 Ais shown in Fig. 15.27.

Figure 15.27

Problem 13. Sketch y=7 cos^22 A between

A= 0 ◦andA= 360 ◦.

Maximum value=7; period= 180 ◦/ 2 = 90 ◦.

A sketch ofy=7 cos^22 Ais shown in Fig. 15.28.