Trigonometric Graphs

Expert reviewed • 08 January 2025 • 8 minute read

HSC Maths Advanced Syllabus

solve trigonometric equations involving functions of the form $kf(a(x+b))+C$ , using technology or otherwise, within a specified domain
use trigonometric functions of the form $kf(a(x+b))+C$ to model and/or solve practical problems involving periodic phenomena

When dealing with the trigonometric graphs, it is important to know their significant features. This involves a complete understanding of the amplitude and period of each graph.

What is Amplitude in Trigonometry?

The amplitude of a trigonometric graph refers to the maximum distance from the graph's midline to its peak or trough. In simpler terms, it is half the distance between the maximum and minimum values of the function.

The basic forms of the trigonometric graphs, each have their own amplitudes.

$sin(x)$ and $cos(x)$ both have an amplitude of 1
$tan(x)$ does not have a specific amplitude, as it infinitely stretches across the y-axis

When vertically dilated, the functions amplitude changes. This is seen in the form:

y=asin(x)\qquad and \qquad y=acos(x)

where the new amplitude of each graph is $a$ .

What is Period in Trigonometry?

The period of a trigonometric graph is the smallest positive value for which the function's values repeat.

The basic forms of the trigonometric graphs, each have their own periods.

$sin(x)$ and $cos(x)$ both have a period of $2\pi$ (full revolution)
$tan(x)$ has a period of $\pi$ (half revolution)

When horizontally dilated the period of a trigonometric graph changes. Determining the period of a graph that has been horizontally dilated is slightly more complicated than determining the amplitude. As such, when a trigonometric function is in the general form:

y=sin(nx+b)

We can determine the period of the graph by dividing the period of the original graph, by the horizontal dilation. For the general graph noted above the calculations would be the following.

period=\frac{2\pi}{n}

Remember that the original period for trigonometric graphs is different Eg. the period for $tan(x)$ is $\pi$ , while the other two graphs have a period of $2\pi$

Practice Question 1

Determine the period and amplitude of the following function. $y=8cos(5x)$ .

To determine the amplitude of the graph, we must apply the value of vertical dilation to that of the original graph.

amplitude=8\times 1\\=8

Now, to determine the period, we must divide the original period for $cos(x)$ by that of the applied horizontal dilation.

period=\frac{2\pi}{5}

What is Phase Shift in Trigonometry?

The phase of a trigonometric graph, refers to how much a trigonometric graph has been horizontally shifted by. This occurs to graphs in the form.

y=sin(cx+\theta)\qquad y=cos(cx+\theta)\qquad y=tan(cx+\theta)

where each trigonometric graph has been shifted along the x-axis toward the left, by a value of $-\frac{\theta}{c}$ .

Similar to other graphs, a function which appears in the form $(x+\theta)$ will be shifted to the left by $\theta$ , while $(x-\theta)$ will be shifted to the right by $\theta$ .

It is imperative to note that when sketching dilations and shifts to a trigonometric graph, all dilations are to be completed before any shifts are applied.

Practice Question 2

State the phase of the function $y=tan(2x-4)$

The phase of the function is calculated by $-\frac{\theta}{c}$

-\frac{\theta}{c}=-\frac{(-4)}{2}\\=\frac{4}{2}\\=2

$\therefore$ The phase of this graph is 2.

Regions of a Plane

Return to Module 2: Graphs and Equations