Like *e*, π is an infinite non-recurring decimal; it starts off as **3.141592653589....** The word **PI** in *NextBASIC* is taken as standing for this number – try **PRINT PI**. ### Trigonometry with SIN, COS, TAN, ASN, ACS and ATN The trigonometrical functions measure what happens when a point moves round a circle. Here is a circle of *radius* **1** (1 what? It doesn't matter, as long as we keep to the same unit all the way through. There is nothing to stop you inventing a new unit of your own for every circle that you happen to be interested in) and a point moving round it. The point started at the *3 o'clock* position, and then moved round in an anti-clockwise direction.  Distance moved\ around circle = a Starting position Radius = 1 *Fig. 6 – Basics of trigonometrical measurements* We have also drawn in, two lines called axes through the centre of the circle. The one through *9 o'clock* and *3 o'clock* is called the *x-axis*, and the one through *6 o'clock* and *12 o'clock* is called the *y-axis*. To specify where the point is, you say how far it has moved round the circle from its *3 o'clock* starting position: let us call this distance *a*. We know that the circumference of the circle is 2π (because its radius is **1** and its diameter is thus **2**): so when it has moved a quarter of the way round the circle, *a*=π/2; when it has moved halfway round, *a*=π; and when it has moved the whole way round, *a*=2π. Given the curved distance round the edge, *a*, two other distances you might like to know are how far the point is to the right of the *y-axis*, and how far it is above the *x-axis*. These are called, respectively, the *cosine* and *sine* of *a*. The functions **COS** and **SIN** on the computer will calculate these. Note that if the point goes to the left of the *y-axis*, then the *cosine* becomes negative; and if the point goes below the *x-axis*, the *sine* becomes negative. Another property is that once a has got up to 2π, the point is back where it started and the *sine* and *cosine* start taking the same values all over again: **SIN (a+2\*PI) = SIN a**\ **COS (a+2\*PI) = COS a** The *tangent* of *a* is defined to be the *sine* divided by the *cosine*; the corresponding function on the computer is called **TAN**. Sometimes we need to work these functions out in reverse, finding the value of *a* that has given *sine, cosine* or *tangent*. The functions to do this are called *arcsine* (**ASN** on the computer), *arccosine* (**ACS**) and *arctangent* (**ATN**).