## Assumed knowledge

The content of the modules:

- Introductory trigonometry (Years 9–10)
- Further trigonometry (Year 10)
- Trigonometric functions (Year 10).

## Motivation

The Greeks developed a precursor to trigonometry, by studying chords in a circle. However, the systematic tabulation of trigonometric ratios was conducted later by the Indian and Arabic mathematicians. It was not until about the 15th century that a 'modern' approach to trigonometry appeared in Europe.

Two of the original motivations for the study of the sides and angles of a triangle were astronomy and navigation. These were of fundamental importance at the time and are still very relevant today.

With the advent of coordinate geometry, it became apparent that the standard trigonometric ratios could be graphed for angles from \(0^\circ\) to \(90^\circ\). Using the coordinates of points on a circle, the trigonometric ratios were extended beyond the angles found in a right-angled triangle. It was then discovered that the trigonometric functions are periodic and so can be used to model periodic behaviour in nature and in science generally.

More recently, the discovery and exploitation of electricity and electromagnetic waves introduced exciting and very powerful new applications of the trigonometric functions. Indeed, of all the applications of classical mathematics, this is possibly one of the most profound and world changing. The trigonometric functions give the key to understanding and using wave motion and manipulating signals in communications. Decomposing signals into combinations of trigonometric functions is known as Fourier analysis.

In this module, we will revise the basics of triangle trigonometry, including the sine and cosine rules, and angles of any magnitude.

We will then look at trigonometric expansions, which will be very important in the later module The calculus of trigonometric functions. All this is best done, initially, using angles measured in degrees, since most students are more comfortable with these units.

We then introduce radian measure, which is a natural way of measuring angles using arc length. This is absolutely essential as a prerequisite for the calculus of the trigonometric functions and also gives us simple formulas for the area of a sector, the area of a segment and arc length of a circle.

Finally, we conclude with a section on graphing the trigonometric functions, which illustrates their wave-like properties and their periodicity.