The circular functions occur throughout mathematics. Fourier showed that, under very general assumptions, an arbitrary function can be decomposed into components each of which is a circular function. The functions get their name from their use in defining a circle in parametric form: if $latex \displaystyle x = a\cos t \qquad\mbox{and}\qquad y = a\sin t … Continue reading Squaring the Circular Functions
Month: December 2015
Factorial 52: A Stirling Problem
How many ways can a deck of cards be arranged? It is very easy to calculate the answer, but very difficult to grasp its significance. There are 52 cards. Thus, the first one may be chosen in 52 ways. The next one can be any of the remaining 51 cards. For the third, there are … Continue reading Factorial 52: A Stirling Problem
How many Christmas Gifts?
We all know the festive carol The Twelve Days of Christmas. Each day, “my true love” receives an increasing number of gifts. On the first day there is one gift, a partridge in a pear tree. On the second, two turtle doves and another partridge, making three. There are six gifts on the third day, … Continue reading How many Christmas Gifts?
The Ping Pong Pendulum
Galileo noticed the regular swinging of a candelabra in the cathedral in Pisa and speculated that the swing period was constant. This led him to use a pendulum to measure intervals of time for his experiments in dynamics. Bu not all pendulums behave like clock pendulums. The Ping Pong Pendulum We consider a pendulum with … Continue reading The Ping Pong Pendulum
The Flight of a Golf Ball
Golf balls fly further today, thanks to new materials and mathematical design. They are a triumph of chemical engineering and aerodynamics. They are also big business, and close to a billion balls are sold every year. [TM081: search for “thatsmaths” at Irish Times ]. The golfer controls the direction and spin of the ball by … Continue reading The Flight of a Golf Ball
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