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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Posted in trigonometry.

Some months ago, I wrote about a method for finding $\cos(72º)$, or $\cos\br{\frac{2\pi}{5}}$ in proper units. Almost immediately, the good people of Twitter and Facebook – notably @ImMisterAl (Al) and @BuryMathsTutor (Mark)- suggested other ways of doing it. Let’s start with Mark’s method, which he dissects in his book GCSE

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Posted in integration, trigonometry.

The ever-challenging Adam Atkinson, having noticed my attention to the “impossible” New Zealand exams, pointed me at a tricky question from an Italian exam which asked students to verify that, to give a smooth ride on a bike with square wheels (of side length 2), the height of the floor

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Posted in ask uncle colin, trigonometry.

Dear Uncle Colin, How would you calculate $\cos(72º)$ by hand? – Pointless Historical Inquiry Hi, PHI, and thanks for your message. There seems to be an awful lot of degree use around at the moment, and I’m not very happy about it. But still, in the spirit of answering what

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Posted in ninja maths, trigonometry.

The Mathematical Ninja sniffed. “$4\sin(15º)$? Degrees? In my classroom?” “Uh uh sorry, sensei, I mean $4\sin\br{\piby{12}}$, obviously, I was just reading from the textmmmff.” “Don’t eat it all at once. Now, $4\sin\br{\piby{12}}$ is an interesting one. You know all about Ailes’ Rectangle, of course, so you know that $\sin\br{\piby{12}}=\frac{\sqrt{6}-\sqrt{2}}{4}$, which

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Posted in ask uncle colin, trigonometry.

Dear Uncle Colin, I have a triangle. All I know is that its angles, $\alpha$, $\beta$ and $\gamma$, satisfy $\cos(\alpha)=\frac{1}{4}$ and $\gamma = 30º$ – and I have to find $\tan(\beta)$. Help! – Can't Obviously See It, Need Explanation Hi, COSINE, and thanks for your message! This is one that

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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 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 dialogue, trigonometry.

“You know how you’re always putting things like ‘just to keep @RealityMinus3 happy’ in your posts?” “Of course, sensei!” “Well… you remember that post about missing solutions in a trig problem?” “Ut-oh.” What follows is a guest post by Elizabeth A. Williams, who is @RealityMinus3 in real life. This thing

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