27.04 Period of sine and cosine functions (Extended)
In this lesson, we will look at an important characteristic of a trigonometric function called it's period. We will also look at horizontal dilations and translations of the basic trigonometric functions. And finally, we will consider trigonometric functions with a combination of different dilations and translations.
Period
The period of a trigonometric function is defined as the time taken to complete one full cycle. For $y=\sin x$y=sinx and $y=\cos x$y=cosx, the period can be measured between successive maximums or minimums, or any other repeated point on the graph with the same function value and characteristics. Analysis of the trigonometric functions $y=\sin x$y=sinx and $y=\cos x$y=cosx reveals a period of $360$360.
Horizontal dilations
A function can be dilated horizontally by multiplying the input by a scale factor. That is, a function $f\left(ax\right)$f(ax) is the function $f\left(x\right)$f(x) horizontally dilated by a factor $\frac{1}{a}$1a.
For trigonometric functions that are horizontally dilated, there will be a change in the period.
Let's consider the graph of $y=\sin2x$y=sin2x sketched together with $y=\sin x$y=sinx:
From the graph, we can see that the function of $y=\sin2x$y=sin2x has a period of $180$180which is half the period of the original graph of $y=\sin x$y=sinx. This dilation has compressed the graph horizontally. It is important to notice that by changing the period of the function the amplitude, domain and range don't change.
Now let's consider the graph of $y=\cos\left(\frac{x}{2}\right)$y=cos(x2) sketched together with $y=\cos x$y=cosx:
From the graph, we can see that the function of $y=\cos\left(\frac{x}{2}\right)$y=cos(x2) has a period of $720$720 which is double the period of the original graph of $y=\cos x$y=cosx. This dilation has stretched the graph horizontally. It is important to notice that by changing the period of the function the amplitude, domain and range don't change.
We can summarise the effects of horizontal dilations on trigonometric graphs as follows:
| Function | Dilation | Period |
|---|---|---|
| $y=\sin\left(ax\right)$y=sin(ax) |
horizontal compression $|a|>1$|a|>1 horizontal stretch $0<|a|<1$0<|a|<1 vertical reflection $a<0$a<0 |
$\frac{360}{a}$360a |
| $y=\cos\left(ax\right)$y=cos(ax) |
horizontal compression $|a|>1$|a|>1 horizontal stretch $0<|a|<1$0<|a|<1 vertical reflection $a<0$a<0 |
$\frac{360}{a}$360a |
Combined translations and dilations of trigonometric functions
As is the case for other functions, we will often want to interpret trigonometric functions that involve multiple translations and dilations. We can combine all the dilations and translations previously mentioned into one equation as follows. Note carefully the structure of the equation and specifically that the input expression to the function is factorised to reveal the values of $a$a and $b$b.
To obtain the graph of $y=kf\left(ax\right)+c$y=kf(ax)+c from the graph of $y=f\left(x\right)$y=f(x):
- $k$k vertically dilates the graph of $y=f\left(x\right)$y=f(x), and corresponds to a change in amplitude
- $a$a horizontally dilates the graph of $y=f\left(x\right)$y=f(x), and corresponds to a change in period
- $c$c vertically translates the graph of $y=f\left(x\right)$y=f(x)
In the case that $k$k is negative, it has the additional property of reflecting the graph of $y=f\left(x\right)$y=f(x) about the horizontal axis.
In the case that $a$a is negative, it has the additional property of reflecting the graph of $y=f\left(x\right)$y=f(x) about the vertical axis.
Try experimenting with the value of each of these variables in the applet below.
Worked examples
Example 1
Sketch the graph of $y=-2\sin3x$y=−2sin3x.
Think: Starting with the graph of $y=\sin x$y=sinx, we can work through a series of transformations so that it coincides with the graph of $y=-2\sin3x$y=−2sin3x.
Do: We can first reflect the graph of $y=\sin x$y=sinx about the $x$x-axis. This is represented by applying a negative sign to the function (multiplying the function by $-1$−1).
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| The graph of $y=-\sin x$y=−sinx |
Then we can increase the amplitude of the function to match. This is represented by multiplying the $y$y-value of every point on $y=-\sin x$y=−sinx by $2$2.
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| The graph of $y=-2\sin x$y=−2sinx |
Next, we can apply the period change that is the result of multiplying the $x$x-value inside the function by $3$3. This means that to get a particular $y$y-value, we can put in an $x$x-value that is $3$3 times smaller than before. Notice that the points on the graph of $y=-2\sin x$y=−2sinx move towards the vertical axis by a factor of $3$3 as a result.
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| The graph of $y=-2\sin3x$y=−2sin3x |