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1. Introduction and the three basic trigonometric functions

Triangles are basic geometric shapes comprising three straight lines, each of which is joined to the two others at its ends. A triangle has three included angles (angles defined by two of the sides) and together these angles sum to 180°. We will start with a special sort of triangle, where one of the included angles is a right-angle (90°). The lengths of the sides of a right-angled triangle can be calculated using Pythagoras' Theorem, and this is described graphically in the Appendix to this document.

Here, the main aim is to introduce a set of relationships between the sides and angles of triangles which form the basis of trigonometry. Most people encounter trigonometric functions in the context of geometry. However, some trigonometric functions are used in other applications, including the construction of regular cycles in models of physical and biological systems.

The three basic trigonometric functions relate an angle to the ratio between pairs of sides of a right-angled triangle. In this diagram, the angle α (Greek 'alpha') is at the junction between the hypotenuse, the longest side of the right-angled triangle, and a side designated the 'adjacent'. The side opposite angle α is designated the 'opposite'. Three ratios between pairs of sides are the sine, cosine and tangent:

Three basic trigonometric functions

You will notice that for the third angle of the triangle, which we can designate β:

β = 90 – α
sine β = cosine α
cosine β = sine α
tangent β = 1/tangent α

Given a value for one of the two non-right angles in a right-angled triangle, and the length of any side, it is possible to calculate the length of either of the other two sides. Conversely, given the length of any two sides, it is possible to calculate the value of either non-right angle from one of the trigonometric functions shown in the figure.

The three functions are usually written in their abbreviated forms, which are 'sin', 'cos' and 'tan'. These terms are used in spreadsheet calculations, for instance as SIN(A12), where the value contained in cell A12 (in brackets) is termed the 'argument' of the function.