### 1 Oscillating functions: amplitude, period and frequency

Particular types of periodic functions ( HELM booklet  2.2) that are especially important in engineering are the sine and cosine functions. These are possible choices when modelling behaviour that involves oscillation or motion in a circle. The usefulness of these functions is rather limited if we confine our attention only to sin $\left(x\right)$ and cos $\left(x\right)$ . Use of functions such as 3sin $\left(2x\right)$ , 5cos $\left(3x\right)$ and so on, and other functions made up of sums of functions of this type, enables the modelling of a great variety of situations where the quantity being modelled is known to change in a periodic way. Here we will examine the behaviour of sine and cosine functions and consider a modelling context where choice of a sine function is appropriate. Figure 6 shows how the terms amplitude , period and frequency are defined with respect to a general sinusoid (the name for any general sine or cosine function).

Figure 6 :

The amplitude represents the difference between the maximum (or minimum) value of a sinusoidal function and its mean value (which is zero in Figure 6). The frequency represents the number of complete cycles of the function in each unit change in $x$ . The period is such that $f\left(x+T\right)=f\left(x\right)$ for all $x$ , e.g. for $sinx$ , $T=2\pi$ .

##### Example 2

Sketch the sinusoids:

1. $y=sinx$
2. $y=2sinx$
3. $y=cosx$
4. $y=cos\frac{x}{2}$
##### Solution

Figure 7

Figure 8

Using the graphs in Figures 7 and 8, state the amplitude, frequency and period of

1. $sinx$
2. $2sinx$
3. $cosx$
4. $cos\frac{x}{2}$

Give frequency and period in terms of $\pi$ .

1. amplitude = 1, frequency = $1∕2\pi$ , period = $2\pi$ .
2. amplitude = 1, frequency = $1∕2\pi$ , period = $2\pi$ .
3. amplitude = 2, frequency = $1∕2\pi$ , period = $2\pi$ .
4. amplitude = 1, frequency = $1∕4\pi$ , period = $4\pi$ .

See Figure 7 for the sine functions and Figure 8 for the cosine functions.

Note that (2) has twice the amplitude of (1) and (4) has half the frequency and twice the period of (3).

Note that the cosine functions $cosnx$ have the same shape as the sine functions $sinnx$ but, at $x=0$ , the cosine functions have a peak or maximum, whereas the sine functions have the value zero, which is the mean value for both of these functions. Indeed the graph of $y=cosx$ is exactly like that for $y=sinx$ with all the $x$ values displaced by $\pi ∕2$ .

More general forms of sine and cosine function are given by $y$ = $asin\left(bx\right)$ , and $y$ = $acos\left(bx\right)$ where $a$ and $b$ are arbitary constants. These are functions with frequency $\frac{b}{2\pi }$ , period $\frac{2\pi }{b}$ and amplitude $a$ . The peak values of the sine functions occur at $x$ values equal to $\frac{\pi }{2}$ , $\frac{5\pi }{2}$ , $\frac{9\pi }{2}$ etc. The minimum values occur at $x$ values equal to $\frac{3\pi }{2}$ , $\frac{7\pi }{2}$ , $\frac{11\pi }{2}$ etc.

When the period is measured in seconds, frequency is measured in cycles per second or Hz which has units of 1/time.

##### Exercises
1. Figure 7 on page 37 shows on the same axes the graphs of $y=sinx$ and $y=2sinx$ .
1. State in words how the graph of $y=2sinx$ relates to the graph of $y=sinx$
2. Sketch the graphs of (i) $y=\frac{1}{2}sinx$ , (ii) $y=\frac{1}{2}sinx+\frac{1}{2}$
2. Figure 8 on page 37 shows on the same axes the graph $y=cosx$ and $y=cos\frac{x}{2}$
1. State in words how the graph of $y=cosx$ relates to the graph of $y=cos\frac{x}{2}$
2. Sketch graphs of (i) $y=cos2x$ , (ii) $y=2cosx$
1. $y=sin2x$ has the same form as $y=sinx$ but all the $y$ values are doubled. The graph is ‘stretched’ vertically.
2. $y=cos\frac{x}{2}$ has the same form as $y=cosx$ but all the $y$ values are halved. The graph is ‘shrunk’ vertically.