Browsing category trigonometry

A MathsJam Masterclass: A Circle That Won’t Behave

Somewhere deep in the recesses of my email folder lurks a puzzle that looks simple enough, but that several of my so-inclined friends haven't found easy: A circle of radius $r$, has centre $C\ (0,r)$. A tangent to the circle touches the axes at $A\ (9,0)$ and $B\ (0, 2r+3)$.

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Ask Uncle Colin: A trigonometric integral

Dear Uncle Colin, I'm trying to find a definite integral: $\int_0^\pi \sin(kx) \sin(mx) \dx$, where $m$ and $k$ are positive integers and the answer needs to be simplified as far as possible. I've wound up with $\left[\frac{ (k+m) \sin((k-m) \pi) - (k-m)\sin((k+m)\pi) }{2(k-m)(k+m)}\right]$, but it's been marked wrong. -- Flat

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A trigonometric coincidence

"Hm," I thought, "that's odd." I don't often work in degrees, but the student's syllabus insisted. And $\sin(50º)$ came up. It's 0.7660, to four decimal places. But... I know that $\sin\left(\frac 13 \pi\right)$, er, sorry, $\sin(60º)$ is 0.8660 -- a difference that's pretty close to $\frac 1{10}$. Which got me

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Ask Uncle Colin: A hellish trigonometric identity

Dear Uncle Colin, @CmonMattTHINK unearthed the challenge to prove that: $\tan\left( \frac 3{11}\pi \right) + 4 \sin\left( \frac 2{11}\pi \right) = \sqrt {11}$. Wolfram Alpha says it's true, but I can barely get started on the proof and I'm worried no-one will like me. Grr, Really Obnoxious Trigonometry Has Evidently

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An interesting GCSE triangle

It's not often I have anything nice to say about EdExcel. I've usually found their exams to be the most predictable and least thought-provoking of all the boards (at least until they finally snapped in 2013 and let Kate the Photographer loose on an unsuspecting cohort). At GCSE, their advanced

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Ask Uncle Colin: A nasty triangle

Ask Uncle Colin is a chance to ask your burning, possibly embarrassing, maths questions -- and to show off your skills at coming up with clever acronyms. Send your questions to and Uncle Colin will do what he can. Dear Uncle Colin, I've been given a trigonometry problem I

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The Mathematical Ninja and Ailles’ Rectangle

"$\sin(15º)$," said the GCSE student, and the Mathematical Ninja -- recognising that the qualification recognised idiotic angle measures -- let it slide. "0.2588", he muttered, under his breath, knowing full well that the exact answer -- $\frac{\sqrt{6} - \sqrt{2}}{4}$ -- would get him a blank stare. He sighed the sigh

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A MathsJam Masterclass

At the East Dorset MathsJam Christmas party, @jussumchick (Jo Sibley in real life) posed the following question: There are two ways to draw a 16-gon with rotational symmetry of order 8 inside a unit circle, as shown. What's the ratio of their areas? Typically, I look at this sort of

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How I approximated $\pi$ for “Pi Day”

There was a post here, but it's not here any more! Instead, it's over at the Aperiodical, as part of their $\pi$ Day approximation challenge.

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Taking Trigonometry Further

On a recent episode of everyone's second-favourite maths podcast, Taking Maths Further, @stecks and @peterrowlett asked: You want to calculate the height of a tall building. You set up a device for measuring angles, on a 1m high tripod, which is 200m away from the building. The angle above horizontal,

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