Posted in Uncategorized

I recently had a flurry of correspondence with translators of The Maths Behind (available wherever etc., but also soon in Swedish and Korean): embarrassingly, they had caught several mistakes in the book. These things happen; we try to put them right and move on. However, it got me wondering: can

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Posted in ask uncle colin, proof, trigonometry

Dear Uncle Colin, I have a trig identity I can't prove! I have to show that $\frac{\cos(x)}{1-\sin(x)} = \tan(x) + \sec(x)$. Strangely Excited Comment About Non-Euclidean Trigonometry. Hi, SECANT, and thanks for your message! This is a slightly sneaky one, but definitely a good one to practice. Let's do it

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Posted in complex numbers

You know how I often bang on about how 'impossible' exams are really nothing of the sort? Well, just for a change, I'm going to bang on about how sometimes exam boards get it wrong. I'm looking at the 2014 Edexcel FP2 paper (the normal one, not the (R) one

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Posted in ask uncle colin, trigonometry

Dear Uncle Colin, I was asked to work out $\tan\br{\theta + \piby 2}$, but the formula failed because $\tan\br{\piby 2}$ is undefined. Is there another way? - Lost Inna Mess, Infinite Trigonometry Hi, LIMIT, and thanks for your message! In fact, there are several ways to approach it! Basic geometry

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Posted in arithmetic, puzzles

An excellent puzzle I heard from @panlepan (I paraphrase, as I've lost the tweet): When you move the final digit of 142857 to the front, you get 714285, which is five times as large. What is the smallest positive integer that is doubled when the last digit moves to the

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Posted in vectors

Dear Uncle Colin, Apparently, the volume of a tetrahedron with three edges given by the vectors $\vec{AB}$, $\vec{AC}$ and $\vec{AD}$, is $\frac{1}{6} \left| \vec{AB} \cdot \br{\vec{AC}\times\vec{AD}} \right|$. Where does that come from? - Very Obviously Lacklustre Understanding of My Exam Hi, VOLUME, and thanks for your message! I think there

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Posted in probability

A nice prompt from @shahlock, some time ago: Math Prompt #apstats #mtbosTwo players A, B. A is 4-0 against B. How would you estimate probability A wins next match? Assume independence — M Shah (@shahlock) November 27, 2016 Stand back, everyone: I'm going to apply Bayes's Theorem. A prior Let's

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Posted in ask uncle colin, trigonometry

Dear Uncle Colin, I'm trying to solve $2\cos(3x)-3\sin(3x)=-1$ (for $0\le \theta \lt 90º$) but I keep getting stuck and/or confused! What do you recommend? - Losing Angles, Getting Ridiculous Answers, Nasty Geometric Equation Hi, LAGRANGE, and thank you for your message! There are a couple of ways to approach this:

Read More →Some time ago, I had a message from someone who - somewhat oddly - wanted to find a centre of rotation (with an unknown angle) without constructing any bisectors. (Obviously, if it was a right-angle rotation, they could use the set-square trick; if it was a half-turn, the centre of

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Posted in trigonometry

Dear Uncle Colin, I'm stuck on a trigonometry question: find $\cos\br{\frac{1}{2}\arcsin\br{\frac{15}{17}}}$. Any bright ideas? - Any Rules Calculating Some Inverse Notation? Hi, ARCSIN, and thanks for your message! That's a nasty one! Let's start by thinking of a triangle with an angle of $\arcsin\br{\frac{15}{17}}$ - the opposite side is 15

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