trigonometry

156 GEOMETRY AND TRIGONOMETRY
Problem 11. Sketch y = 2 cos(ωt − 3π/10)
over one cycle.
Amplitude = 2; period = 2π/ω rad.
2 cos(ωt − 3π/10) lags 2 cos ωt by 3π/10ω seconds.
A sketch of y = 2 cos(ωt − 3π/10) is shown in
Fig. 15.24.
Figure 15.25
Figure 15.26
Figure 15.24
Graphs of sin 2 A and cos 2 A
(i) A graph of y = sin 2 A is shown in Fig. 15.25
using the following table of values.
A ◦ sin A (sin A) 2 = sin 2 A
0 0 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 of y = cos 2 A is 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 ◦ (or π rad) and both
contain only positive values. Thus a graph of
y = sin 2 2A has a period 180 ◦ /2, i.e., 90 ◦ . Similarly,
a graph of y = 4 cos 2 3A has a maximum
value of 4 and a period of 180 ◦ /3, i.e. 60 ◦ .
Problem 12.
0 < A < 360 ◦ .
Sketch y = 3 sin 2 2 1 A in the range
Maximum value = 3; period = 180 ◦ /(1/2) = 360 ◦ .
A sketch of 3 sin 2 2 1 A is shown in Fig. 15.27.
Figure 15.27
Problem 13. Sketch y = 7 cos 2 2A between
A = 0 ◦ and A = 360 ◦ .
Maximum value = 7; period = 180 ◦ /2 = 90 ◦ .
A sketch of y = 7 cos 2 2A is shown in Fig. 15.28.