### Archive

Posts Tagged ‘calculus’

## Trigonometry

I have just realized that I haven’t posted a good puzzle here in a long time, so here it goes one on Trigonometry, that the average student of Calculus should be able to tackle: you can use anything you think it could help: derivatives, symmetry, periodicity, integration, summation, go to several variables, differential equations, etc

Prove that, for all real values $x \in \mathbb{R},$ it is

$\sin x = x \cos\big(\tfrac{x}{2}\big) \cos\big(\tfrac{x}{4}\big) \cos\big(\tfrac{x}{6}\big) \cos\big(\tfrac{x}{8}\big) \dotsb \cos\big( \tfrac{x}{2n}\big) \dotsb$

or, in a more compact notation,

$\sin x = x \displaystyle{\prod_{n=1}^\infty \cos \big( \tfrac{x}{2n} \big)}$

## So you want to be an Applied Mathematician

My soon-to-be-converted Algebrist friend challenged me—not without a hint of smugness in his voice—to illustrate what was my last project at that time. This was one revolving around the idea of frames (think of it as redundant bases if you please), and needed proving a couple of inequalities involving sequences of functions in $L_p$—spaces, which we attacked using a beautiful technique: Bellman functions. About ninety minutes later he conceded defeat in front of the board where the math was displayed. He promptly admitted that this was no Fortran code, and showed a newfound respect and reverence for the trade.