Amplitude of sine and cosine

The graphs of trigonometric functions like $y=\sin x$y=sinx and $y=\cos x$y=cosx have sections that are repeated in the $x$x-direction. The length of these repeated sections is called the wavelength, and we can understand a lot about the behaviour of the whole function by looking only at what happens within a single wavelength.

Typically, each trigonometric graph will have a minimum value, a maximum value, and average value along the wavelength. The distance between the maximum (or minimum) value and the average value is called the amplitude.

If the range of the function is given by $\text{maximum value - minimum value }$maximum value - minimum value , the amplitude is then $\frac{\text{max - min }}{2}$max - min 2​. Notice that the amplitude is always a positive number.

Dilations and the amplitude

An equation of the form $y=a\sin x$y=asinx has an amplitude of $a$a units. If we start with the equation $y=\sin x$y=sinx, where $a=1$a=1, we can transform this equation by changing the amplitude.

Graphically this transformation corresponds to stretching the graph of $\sin x$sinx in the vertical direction when $a>1$a>1, and compressing the graph of $\sin x$sinx in the vertical direction when $00<a<1. We can also reflect the graph across the $x$x-axis by multiplying the whole function by $-1$−1.

This action of stretching and compressing the graph of a function is known as dilation. For trigonometric graphs like $\sin x$sinx and $\cos x$cosx, a vertical dilation by a factor of $a$a is equivalent to increasing the amplitude by a factor of $a$a.

 

Worked example 1

State the amplitude of the function $f\left(x\right)=5\sin x$f(x)=5sinx.

Think: When we compare $f\left(x\right)$f(x) to the standard function $\sin x$sinx we can see that all the function values of $5\sin x$5sinx will be five times larger than all the function values of $\sin x$sinx. This means that the amplitude of $5\sin x$5sinx is also five times larger than the amplitude of $\sin x$sinx.

Do: The amplitude of $\sin x$sinx is $1$1, so the amplitude of $f\left(x\right)=5\sin x$f(x)=5sinx is $5\times1=5$5×1=5.

Reflect: We can obtain the graph of $f\left(x\right)=5\sin x$f(x)=5sinx by starting with the graph of $y=\sin x$y=sinx and applying a vertical dilation by a factor of $5$5.

 

Worked example 2

The graph of $y=\cos x$y=cosx is reflected across the $x$x-axis, then compressed in the vertical direction so that its minimum value is $\frac{3}{4}$34​. What is the equation of the resulting function. What is the amplitude?

Think: Recall that a reflection across the $x$x-axis corresponds to multiplying the function by $-1$−1. This will "switch" the location of the maximum and minimum values of the graph, but the amplitude will still be a positive value.

Do: Let's keep track of how the equation of the graph changes at each stage of the transformation.

$y=\cos x$y=cosx	$\rightarrow$→	$y=-\cos x$y=−cosx	(Reflection across $x$x-axis)
$y=-\cos x$y=−cosx	$\rightarrow$→	$y=-\frac{3}{4}\cos x$y=−34​cosx	(Vertical compression)

The final equation of the resulting graph is $y=-\frac{3}{4}\cos x$y=−34​cosx. The amplitude of this equation is $\frac{3}{4}$34​.

Reflect: Compare the resulting equation with the original equation. The only difference is the constant multiple of $-\frac{3}{4}$−34​. Notice that although this number is negative, the amplitude of the resulting equation is positive. In general, the amplitude of the equation $y=a\cos x$y=acosx is $\left|a\right|$|a|.

 

Practice questions

Question 1

Question 2

Question 3