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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Posted in ask uncle colin, trigonometry.

Dear Uncle Colin, When I solve $2\tan(2x)-2\cot(x)=0$ (for $0 \le x \le 2\pi$) by keeping everything in terms of $\tan$, I get four solutions; if I use sines and cosines, I get six (which Desmos agrees with). What am I missing? – Trigonometric Answers Not Generated – Expecting 'Nother Two

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Posted in ninja maths, trigonometry.

As the student was wont to do, he idly muttered "So, that's $\cos(10º)$…" The calculator, as calculators are wont to do when the Mathematical Ninja is around, suddenly went up in smoke. "0.985," with a heavy implication of 'you don't need a calculator for that'. As the student was wont

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Posted in ask uncle colin, trigonometry.

Dear Uncle Colin, In an answer sheet, they've made a leap from $\arctan\left(\frac{\cos(x)+\sin(x)}{\cos(x)-\sin(x)}\right)$ to $x + \frac{\pi}{4}$ and I don't understand where it's come from. Can you help? — Awful Ratio Converted To A Number Hello, ARCTAN, and thank you for your message! There's a principle I want to introduce

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