Over the years we have come across many different methods of proving the sine function addition formula

and we have long felt it could be both interesting and useful to gather together in one article a list of methods to illustrate how different ideas from geometry, algebra and calculus may be used to prove it.

In the elementary theory of collisions of two small particles, it is a well-known result that the momentum of the system as a whole is the same after impact as before. This is because the impulses on the two particles are taken to be equal and opposite, thereby implying zero total impulse on the system. It is not obvious that the same will be true after rolling snooker balls collide because, apart from the impulsive forces due to the impact, frictional forces also act on the balls. These are caused by the friction of the table when the balls are in a temporary sliding phase immediately following the impact.

In his recent note “On buying postage stamps” in the June 1987 Gazette, John Gilder discusses the solution in integers of the equation

where x is the number of second class stamps (at the old rate of 12p), y the number of first class stamps (at 17p), and z the total cost in pounds. A simple algorithm, which can also be applied to equations involving three or more independent variables, is obtained from the following result.

The Entrepreneur competition in the Sunday Times (and the Portfolio competition in the Times) have the following structure. A list of (say 10) integers a(1),a(2),…,a(10) is given in the paper (in the form of share prices) together with a target integer T. Readers have a card on which is printed a second list of (say 4) integers b(1),b(2),b(3),b(4), this second list being different for each reader.

In exploring the probabilities involved in a roll-a-penny game, some properties turned out to be a little more complicated than expected. The main aim here is to find the probability that the player will lose after a certain number of rolls.

Suppose we have a board upon which squares are drawn, and that the side of each square is four times the radius of a penny. (In practice, this means that the squares are of side 4 cm.) The first question is, “What is the probability that the penny lands entirely within a square, assuming that the dividing lines are of negligible thickness?”

The mensuration formulae for curved objects ought to be one of the highlights of everyone’s elementary mathematical education. I get my pupils at 13 and they've already been taught that the area of a circle is πr2. It is very upsetting to find that they usually have seen no sort of justification of this result, so they don’t think of it as surprising when π, which was introduced in the circumference formula, mysteriously pops up again in the area formula.

This year I thought I would put some effort into justifying the surface area formula for the sphere. I don’t claim that what follows is very original, and of course the main idea goes back to Archimedes, but I hope this article will save readers a bit of work at least.

Let k be a positive integer and a > 1. It is well known that

.

The shortest proof seems to be the following. By estimating the exponential function with only one term from its series we see at once that

.

In “The music of the spheres”, Jacob Bronowski takes us on a mathematical tour of the Alhambra in Grenada. This includes a visit to the Sala de las Camas, that most private inner sanctum, in order to study the tessellations on the walls. We have used this film with teachers in training at Leeds University for some years now. It provides a stimulating introduction to the history of mathematics and enables consideration of the place of historical references in the teaching of mathematics in schools. We have also compiled a worksheet, based on the film, which includes the task of constructing one of the tessellations, illustrated overleaf.

Suppose we are given a sequence of numbers defined by a linear recurrence relation with constant coefficients, for example the Fibonacci numbers which are defined by:

It is a well-known fact that the number of hours of daylight varies with the time of year, except at points on the equator. It is sometimes thought that at latitudes between the Arctic and Antarctic circles the variation takes the form of a sine curve. However there is no theoretical basis for this idea, although it is true that a sine curve may give quite a good approximation.