Posted in ask uncle colin, proof, trigonometry

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

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Posted in geometry

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

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Posted in ask uncle colin, trigonometry

Dear Uncle Colin, I'm normally pretty good at simultaneous equations, but I can't figure out how to solve this for $a$ and $b$. $\cos(a)-\cos(b) = x$ $\sin(a)-\sin(b) = y$ - Any Random Circle Hi, ARC, and thanks for your message! This is, it turns out, a bit trickier than it

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Posted in trigonometry

What's that, @pickover? Shiver in ecstasy, you say? Just for a change. Shiver in ecstasy. The sides of a pentagon, hexagon, & decagon, inscribed in congruent circles, form [a] right triangle. pic.twitter.com/Uastgc7SJo — Cliff Pickover (@pickover) May 20, 2017 That's neat. But why? Let's suppose the circles all have radius

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Posted in geometry, ninja maths, quadratics

Dear Uncle Colin, I've got a funny square and I can't find $x$. Can you help? - Oughta Be Simple, Can't Unravel Resulting Equations Hi, OBSCURE, and thanks for your message! You're right, it ought to be simple... but it turns out not to be. It is simple enough to

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Posted in algebra

This puzzle presumably came to me by way of @ajk44, some time ago. Thanks, Alison! The problem, given here, is to find the equations of two lines that complete a square, given: Two of the lines are $y=ax+b$ and $y=ax+c$ One of the vertices is at $(0,b)$. The example given

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Posted in ask uncle colin

Dear Uncle Colin, I'm told that the graphs of the functions $f(x) = x^3 + (a+b)x^2 + 3x - 4$ and $g(x) = (x-3)^3 + 1$ touch, and I have to determine $a$ in terms of $b$. Where would I even start? - Touching A New Graph Except Numerically Troubling

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Posted in integration

Some while ago, I showed a slightly dicey proof of the Basel Problem identity, $\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac {\pi^2}{6}$, and invited readers to share other proofs with me. My old friend Jean Reinaud stepped up to the mark with an exercise from his undergraduate textbook: The French isn't that difficult,

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Posted in ask uncle colin

Dear Uncle Colin I've been asked to find $\sum_3^\infty \frac{1}{n^2-4}$. Obviously, I can split that into partial fractions, but then I get two series that diverge! What do I do? - Which Absolute Losers Like Infinite Series? Hi, WALLIS, and thanks for your message! Hey! I'm an absolute loser who

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Posted in podcasts

In this month's episode of Wrong, But Useful, Colin and Dave are joined by @niveknosdunk, who is Professor Kevin Knudson in real life. Kevin, along with previous Special Guest Co-Host @evelynjlamb, has recently launched a podcast, My Favorite Theorem The number of the podcast is 12; Kevin introduces us to

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