Ask Uncle Colin: A Cosec Proof | Colin

Dear Uncle Colin

I’m stuck on a trigonometry proof: I need to show that $\cosec(x) – \sin(x) \ge 0$ for $0 < x < \pi$. How would you go about it?

– Coming Out Short of Expected Conclusion

Hi, COSEC, and thank you for your message! As is so often the case, there are several ways to approach this.
The most obvious one

The first approach I would try would be to turn the left hand side into a single fraction: $\frac{1}{\sin(x)} – \sin(x) \equiv \frac{1 – \sin^2(x)}{\sin(x)}$.

The top of that is $\cos^2(x)$, so you have $\frac{\cos^2(x)}{\sin(x)}$.

In the specified region,…

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Life in the sixth form funding canyon | Eddie Playfair

The chancellor is currently pondering his priorities for the forthcoming autumn statement on public spending and given the critical state of our public services there are plenty of worthy calls on resources. Without minimising the case for spending more on health, housing, social care and benefits, it’s also worth reminding the chancellor of the pressing … Continue reading Life in the sixth form funding canyon

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Standard Non-Conformity | Sam Shepherd

As I’ve blogged about before, language standards are somewhat problematic: my definition of right can vary significantly from your definition of right, and the term “standard” is very often a lazy discriminatory euphemism for prestige: essentially “if you don’t speak like the rich and powerful, then there’s something wrong with you.” Language is a bugger to control, particularly when the standard being sold is fairly arbitrary, like the finally disappearing rule of not splitting infinitives. To dictate a specific form requires a belief in absolute definitions, incontrovertible rules, even…

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Practical Tips to Help Bridge the Gender Gap: For Men AND Women in the Workplace @WomenEd_UAE | @DrCWebbBAPhD

I’m inspired to write this based on an online newspaper article I read 2 months ago about a female, Emirati mechanical engineer who was making a plea for change in male dominated industries (see full article at http://ift.tt/2ihpuvx).

My starting point is to say that the WEF Global Gender Gap Report (2016) makes it abundantly clear that men and women and the whole of human society all stand to gain by achieving a Planet5050 and full gender parity in health, economics, education…

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Standard Non-Conformity | Sam Shepherd

As I’ve blogged about before, language standards are somewhat problematic: my definition of right can vary significantly from your definition of right, and the term “standard” is very often a lazy discriminatory euphemism for prestige: essentially “if you don’t speak like the rich and powerful, then there’s something wrong with you.” Language is a bugger to control, particularly when the standard being sold is fairly arbitrary, like the finally disappearing rule of not splitting infinitives. To dictate a specific form requires a belief in absolute definitions, incontrovertible rules, even…

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Tessellations and cuboids | Colin

On a recent1 episode of Wrong, But Useful, Dave mentioned something interesting2: if you take three regular shapes that meet neatly at a point – for example, three hexagons, or a square and two octagons – and make a cuboid whose edges are in the same ratio as the number of sides on each shape (e.g., 6 by 6 by 6 or 4 by 8 by 8), the resulting cuboid has a volume that’s (numerically) the same as its surface area (here, 216 or 256).
Let’s prove it!

The interior angle of a regular $n$-gon is $\pi – \frac{2n}{\pi}$, or $180 – \frac{360}{n}$ if you insist on silly angle measures. In fact, it’s…

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