Browsing category trigonometry

How the Mathematical Ninja approximates $\sin(55º)$

"$0.819$," said the Mathematical Ninja, in as weary a voice as the student used to say "I suppose you're going to tell me how." The nunchaku looked a little rusty, and the axe was in need of a good sharpening. The throwing knives could have done with a clear, and

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Getting closer to $\pi$

A lovely curiosity came my way via @mikeandallie and @divbyzero: In 1992 Daniel Shanks observed that if p~pi to n digits, then p+sin(p)~pi to 3n digits. For instance, 3.14+sin(3.14)=3.1415926529… — Dave Richeson (@divbyzero) July 15, 2016 Isn't that neat? If I use an estimate $p = 3.142$, then this method

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Some interesting confusion

I have a tendency to write about interesting questions from a ‘here’s how you do it’ point of view, which must give the impression that I never get confused1. To try to dispel that, I wanted to share something that came up in an Oxford entrance paper (the MAT from

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The Mathematical Ninja and the Tangents Near 1

“Forty-two degrees,” said the Mathematical Ninja, as smugly as possible while still using degrees. The student’s hand had barely twitched towards the calculator. “Go ahead, punk,” said the Mathematical Ninja. “Make my day.” “Righto,” said the student, and tapped in $\tan^{-1} \left( 0.9 \right)$, carefully closing the bracket. “41.987. That’s

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Ask Uncle Colin: Rational Trigonometric Values

Dear Uncle Colin, You know how sometimes $\sin(2x)$ is rational and $\sin(5x)$ is rational and $\sin(7x)$ is rational, right? Would that necessarily mean that $\sin(12x)$ is rational? Asking for a friend. — Perhaps You THink All Geometry’s On Right Angled Stuff Hi, PYTHAGORAS, I believe it does! (In fact, I

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An alternative proof of the $\sin(2x)$ identity

Uncle Colin recently explained how he would prove the identity $\sin(2x) \equiv 2 \sin(x)\cos(x)$. Naturally, that isn't the only proof. @traumath pointed me at an especially elegant one involving the unit circle. Suppose we have an isosceles triangle set up like this: The vertical 'base' of the triangle is $2\sin(\alpha)$

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Ask Uncle Colin: What is $\tan(1º)$?

Dear Uncle Colin, In my trigonometry homework, I've been asked to find the exact value of $\tan(1º)$. I have no idea where to start. Help! -- A Nice, Gentle Little Exercise Hello, ANGLE. That's an evil question, on many levels. It's doable, but it's not at all easy. In this

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A curious identity

There's something neat about an identity or result that seems completely unexpected, and this one is an especially nice one: $$ e^{2\pi \sin \left( i \ln(\phi)\right) }= -1$$ (where $\phi$ is the golden ratio.) It's one of those that just begs, "prove me!" So, here goes! I'd start with the

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A lovely trigonometric identity

When I was researching the recent Mathematical Ninja piece for Relatively Prime, I stumbled on something at MathWorld1 I'd never noticed before: if $t = \tan(x)$, then: $\tan(2x) = \frac{2t}{1 - t^2}$ $\tan(3x) = \frac{3t - t^3}{1 - 3t^2}$ $\tan(4x) = \frac{4t - 4t^3}{1 - 6t^2 + t^4}$ $\tan(5x) =

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Ask Uncle Colin: Two Answers For One Triangle

Dear Uncle Colin, I have a triangle with sides 4.35cm, 8cm and 12cm; the angle opposite the 4.35cm side is 10º1 and need to find the largest angle. I know how to work this out in two ways: I can use the cosine rule with the three sides, which gives

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