5.2 Properties of Graphs of Trigonometric Functions

We saw in Section 5.1 how the graphs of the trigonometric functions repeat every \(2\pi\) radians. In this section we will discuss this and other properties of graphs, especially for the sinusoidal functions (sine and cosine).

First, recall that the domain of a function \(f(x)\) is the set of all numbers \(x\) for which the function is defined. For example, the domain of \(f(x) = \sin\;x\) is the set of all real numbers, whereas the domain of \(f(x) = \tan\;x\) is the set of all real numbers except \(x=\pm\,\frac{\pi}{2}\), \(\pm\,\frac{3\pi}{2}\), \(\pm\,\frac{5\pi}{2}\), \(...\). The range of a function \(f(x)\) is the set of all values that \(f(x)\) can take over its domain. For example, the range of \(f(x)=\sin\;x\) is the set of all real numbers between \(-1\) and \(1\) (i.e. the interval \(\ival{-1}{1}\)), whereas the range of \(f(x) = \tan\;x\) is the set of all real numbers, as we can see from their graphs.

A function \(f(x)\) is periodic if there exists a number \(p>0\) such that \(x+p\) is in the domain of \(f(x)\) whenever \(x\) is, and if the following relation holds:

There could be many numbers \(p\) that satisfy the above requirements. If there is a smallest such number \(p\), then we call that number the period of the function \(f(x)\).

The above example made use of the graph of \(\sin\;2x\), but the period can be found analytically. Since \(\sin\;x\) has period \(2\pi\),^[1] we know that \(\sin\;(x+2\pi) = \sin\;x\) for all \(x\). Since \(2x\) is a number for all \(x\), this means in particular that \(\sin\;(2x+2\pi) = \sin\;2x\) for all \(x\). Now define \(f(x)=\sin\;2x\). Then

for all \(x\), so the period \(p\) of \(\sin\;2x\) is at most \(\pi\), by our definition of period. We have to show that \(p>0\) can not be smaller than \(\pi\). To do this, we will use a proof by contradiction. That is, assume that \(0<p<\pi\), then show that this leads to some contradiction, and hence can not be true. So suppose \(0<p<\pi\). Then \(0<2p<2\pi\), and hence

for all \(x\). Since any number \(u\) can be written as \(2x\) for some \(x\) (i.e \(u = 2(u/2)\)), this means that \(\sin\;u = \sin\;(u+2p)\) for all real numbers \(u\), and hence the period of \(\sin\;x\) is as most \(2p\). This is a contradiction. Why? Because the period of \(\sin\;x\) is \(2\pi > 2p\). Hence, the period \(p\) of \(\sin\;2x\) can not be less than \(\pi\), so the period must equal \(\pi\).

The above may seem like a lot of work to prove something that was visually obvious from the graph (and intuitively obvious by the “twice as fast” idea). Luckily, we do not need to go through all that work for each function, since a similar argument works when \(\sin\;2x\) is replaced by \(\sin\;\omega x\) for any positive real number \(\omega\): instead of dividing \(2\pi\) by \(2\) to get the period, divide by \(\omega\). And the argument works for the other trigonometric functions as well. Thus, we get:

If \(\omega < 0\), then use \(\sin\;(-A) = -\sin\;A\) and \(\cos\;(-A) = \cos\;A\) (e.g. \(\sin\;(-3x) = -\sin\;3x\)).

We know that \(\;-1 \le \sin\;x \le 1\;\) and \(\;-1 \le \cos\;x \le 1\;\) for all \(x\). Thus, for a constant \(A \ne 0\),

for all \(x\). In this case, we call \(\abs{A}\) the amplitude of the functions \(y=A\,\sin\;x\) and \(y=A\,\cos\;x\). In general, the amplitude of a periodic curve \(f(x)\) is half the difference of the largest and smallest values that \(f(x)\) can take:

In other words, the amplitude is the distance from either the top or bottom of the curve to the horizontal line that divides the curve in half, as in Figure 5.2.3.

Not all periodic curves have an amplitude. For example, \(\tan\;x\) has neither a maximum nor a minimum, so its amplitude is undefined. Likewise, \(\cot\;x\), \(\csc\;x\), and \(\sec\;x\) do not have an amplitude. Since the amplitude involves vertical distances, it has no effect on the period of a function, and vice versa.

So far in our examples we have been able to determine the amplitudes of sinusoidal curves fairly easily. This will not always be the case.

In general, a combination of sines and cosines will have a period equal to the lowest common multiple of the periods of the sines and cosines being added. In Example 5.9, \(\sin\;x\) and \(\cos\;x\) each have period \(2\pi\), so the lowest common multiple (which is always an integer multiple) is \(1 \,\cdot\, 2\pi = 2\pi\).

Generalizing Example 5.9, an expression of the form \(a\,\sin\;\omega x \;+\; b\,\cos\;\omega x\) is equivalent to \(\sqrt{a^2 + b^2}\;\sin\;(x+\theta)\), where \(\theta\) is an angle such that \(\cos\;\theta = \frac{a}{\sqrt{a^2 + b^2}}\) and \(\sin\;\theta = \frac{b}{\sqrt{a^2 + b^2}}\). So \(y=a\,\sin\;\omega x \;+\; b\,\cos\;\omega x\) will have amplitude \(\sqrt{a^2 + b^2}\). Note that this method only works when the angle \(\omega x\) is the same in both the sine and cosine terms.

We have seen how adding a constant to a function shifts the entire graph vertically. We will now see how to shift the entire graph of a periodic curve horizontally.

Consider a function of the form \(y=A\,\sin\;\omega x\), where \(A\) and \(\omega\) are nonzero constants. For simplicity we will assume that \(A >0\) and \(\omega > 0\) (in general either one could be negative). Then the amplitude is \(A\) and the period is \(\frac{2\pi}{\omega}\). The graph is shown in Figure 5.2.10.

Now consider the function \(y=A\,\sin\;(\omega x - \phi)\), where \(\phi\) is some constant. The amplitude is still \(A\), and the period is still \(\frac{2\pi}{\omega}\), since \(\omega x - \phi\) is a linear function of \(x\). Also, we know that the sine function goes through an entire cycle when its angle goes from \(0\) to \(2\pi\). Here, we are taking the sine of the angle \(\omega x - \phi\). So as \(\omega x - \phi\) goes from \(0\) to \(2\pi\), an entire cycle of the function \(y=A\,\sin\;(\omega x - \phi)\) will be traced out. That cycle starts when

Thus, the graph of \(y=A\,\sin\;(\omega x - \phi)\) is just the graph of \(y=A\,\sin\;\omega x\) shifted horizontally by \(\frac{\phi}{\omega}\), as in Figure 5.2.11. The graph is shifted to the right when \(\phi >0\), and to the left when \(\phi <0\). The amount \(\frac{\phi}{\omega}\) of the shift is called the phase shift of the graph.

The phase shift is defined similarly for the other trigonometric functions.

In engineering two periodic functions with the same period are said to be out of phase if their phase shifts differ. For example, \(\sin\;\left( x - \frac{\pi}{6} \right)\) and \(\sin\;x\) would be \(\frac{\pi}{6}\) radians (or \(30\Degrees\)) out of phase, and \(\sin\;x\) would be said to lag \(\sin\;\left( x - \frac{\pi}{6} \right)\) by \(\frac{\pi}{6}\) radians, while \(\sin\;\left( x - \frac{\pi}{6} \right)\) leads \(\sin\;x\) by \(\frac{\pi}{6}\) radians. Periodic functions with the same period and the same phase shift are in phase.

The following is a summary of the properties of trigonometric graphs: