# Graphs of the Trigonometric Functions

By M Bourne

Where do we use this?

In the UK, electricity is supplied at 240 V, 50 Hz. The graph of this voltage has a period of 0.02 sec and an amplitude of 240 V.

### Tip

The best thing to do in this section is to learn the basic shapes of each graph. Then it is only a matter of considering what effect the variables *a*, *b* and *c* are having. You can do a table of values and join the dots, but that becomes painful very quickly.

### Trigonometry Applications

Check out lots more uses of trigonometry.

## Why study trigonometric graphs?

The graphs in this section are probably the most commonly used in all areas of science and engineering. They are used for **modelling** many different natural and mechanical phenomena (populations, waves, engines, acoustics, electronics, UV intensity, growth of plants and animals, etc).

The trigonometric graphs in this chapter are **periodic**, which means the shape repeats itself exactly after a certain amount of time. Anything that has a **regular cycle** (like the tides, temperatures, rotation of the earth, etc) can be modelled using a sine or cosine curve.

## In this chapter...

1. Graphs of y = a sin x and y = a cos x, talks about **amplitude.** Amplitude is a indication of how much energy a wave contains.

2. Graphs of y = a sin bx and y = a cos bx introduces the **period** of a trigonometric graph.

3. Graphs of y = a sin(bx + c) and y = a cos(bx + c) helps you to understand the **displacement** (or **phase shift**) of a trigonometric curve.

4 Graphs of tan x, cot x, sec x and csc x are not as commonly used in the study of periodic activity, but are used in some applications.

5. Applications of Trigonometric Graphs includes the interesting What are the frequencies of music notes?.

6. Composite Trigonometric Curves arise when we add more than one waveform.

7. Lissajous Figures are a special kind of composite trigonometric graph.

## Overview of Trigonometric Graphs

Here's a movie that gives an overview of the concepts in this chapter, by highlighting the various animations you'll find.

We begin the chapter with an examination of what amplitude means and the effect of the "`a`" variable in 1. Graphs of y = a sin x and y = a cos x »