5.5 Graphs, amplitude and period
The graphs of \(\sin \) and \(\cos \) are waves of height \(1\) repeating every \(360^\circ \); \(\tan \) behaves differently, repeating every \(180^\circ \) with vertical asymptotes.
Definition 5.17. For \(y=a\sin (b\theta )\) or \(y=a\cos (b\theta )\):
- (i).
- the amplitude is \(|a|\), half the distance from lowest to highest;
- (ii).
- the period is \(\frac {360^\circ }{b}\), the horizontal length of one complete cycle.
Example 5.18. State the amplitude and period of \(y=3\sin 2\theta \), and of \(y=\frac {1}{2}\cos \frac {\theta }{3}\).
Solution. \(y=3\sin 2\theta \). Here \(a=3\) and \(b=2\): \[\text {amplitude}=3,\qquad \text {period}=\frac {360^\circ }{2}=180^\circ .\] The wave is three times as tall and repeats twice as often as \(\sin \theta \).
\(y=\frac {1}{2}\cos \frac {\theta }{3}\). Here \(a=\frac {1}{2}\) and \(b=\frac {1}{3}\): \[\text {amplitude}=\frac {1}{2},\qquad \text {period}=\frac {360^\circ }{\frac {1}{3}}=1080^\circ .\]
Note 5.19. A larger \(b\) gives a shorter period, because \(b\) divides. It is natural to expect the opposite, so check the direction of the effect against the formula rather than against intuition.
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