# maths

## t-Formulae and parameterisation of the circle

t-Formulae are used in integration to tackle rational expressions of tigonometric functions.  After a spell in the cold, when they were not included in some A level specifications, they are now back in sixth form lessons. It all starts with the subsitution, from which the following functions can be derived, ,  ,  . These derivations …

## Rabbits, Matrices and the Golden Section – nth term of the Fibonacci Sequence by diagonalising a matrix

Fibonacci Sequence Leonardo Pisano, or Leonardo Fibonacci, studied rabbit populations in 1202 in the following way. Rabbit couples (male and female) inhabit an island.  Each rabbit couple becomes fertile 2 months after being born and then begets a male-female pair every month thereafter.  If the population of the island starts with one couple, how many …

## Area of a circle – proof by exhuastion

Once has been defined as the ratio circumference to diameter, the area of a circle must be . A proof relies on an infinite, limiting process which paves the way to some calculus-like ideas. A circle is cut up into 6 sectors which are then rearranged into a near rectangle, What doesn’t look too close …

## Carr’s Synopsis and Sixth Term Entrance Papers

Students in search of extension work in maths, particularly as far as Sixth Term Entrance Papers (STEP) are concerned, should peruse Carr’s Synopsis of results in Elementary Pure Mathematics. This book is famous for being the volume used by the autodidact Ramanujan to teach himself mathematics.  The book is merely a list of results and …

## Transformations of Sine

Functional notation and transformations is always tricky to teach and understand.  GCSE students will meet this in Year 11.   In general, transformations applied after the function are more easily understood:  or   Something very un-intuitive happens when the transformation is applied to the argument of the function:  or   with things stretching when they …

## The Ghosts of Departed Quantities and the difficulties of teaching and learning caculus

Bishop Berkeley writes this attack on the apparent supernatural reasoning involved in calculus.  The infidel was probably Halley (of comet fame) or Newton. If pupils find the subject difficult to understand at school, and teachers find it difficult to teach, then the reason may be articulated in this book by the great man. Quotes include: …

## Euler and the Properties of the Second Derivative

The summer examination season sees pupils searching for maximums and minimums on their text papers. This long-standing pursuit was initiated by the likes of Newton and Leibniz in their calculus. It can be all too difficult to think about for some: “our modern Analysts are not content to consider only the Differences of finite Quantities: …

## Yachting and similar triangles

A yacht has two parallel masts which are both perpendicular to a level deck.  One mast is 12 m high, the other is 8 m.  Stays are rigged from the head if each mast to the foot of the other.  Find the perpendicular height of the crossing point above the deck. Finding the height of …

## The linear combination of a sine and a cosine is itself a sine wave

A linear combination of two functions, and is a sum involving constant multiples of the functions.  That is, where . So, in the case of and , we would have, . It is a slightly surprising fact that the linear combination of two sine waves is itself a sine wave.  The set of sine waves …