Cosines of small angles – Secrets of the Mathematical Ninja

"Where the hell have you been?" asked the student.

The Mathematical Ninja raised an eyebrow into his well-tanned forehead. That, he didn't say, would be telling.

The student sighed and sketched out a triangle. "I know that doesn't look like 2º," she said, to forestall any criticism.

The Mathematical Ninja nodded: "It's a sketch. The details don't matter."

"The hypotenuse is 100 metres," she said, "and I want the... adjacent side. Ah, rubbish. I could do it with the opposite - that would be about three metres, right?"

"Three and half or so, yep," said the Mathematical Ninja. "But you can work out $\cos(x)$ for small angles, too. It's a bit less than 1, generally, but more precisely, it's $\cos(x) \simeq 1 - \frac{x^2}{2}$."

"Where does that come from?"

"Euler series," said the Mathematical Ninja. "Alternatively, you can say $\sin(x) \simeq x$ and use the binomial expansion on $(1 - \sin^2(x))^{1/2})$."

"I'll take your word for it," she said. "So, to get $\cos(x)$, I'm going to need to convert to radians, square, halve, and take away from one? Sounds like a lot of work."

"It's not trivial," admitted the Mathematical Ninja, "but none of those things are too difficult."

The student narrowed her eyes. "Right," she said. "Two degrees is about $\frac{7}{200}$, which squares to $\frac{49}{40,000}$ - that's ridiculous, isn't it? Wait, I can round that to $\frac{50}{40,000}$, which cancels to $\frac{1}{800}$. Halve it, that's $\frac{1}{1,600}$, which is... argh! about $\frac{6}{10,000}$?"

The Mathematical Ninja nodded. "Keep going!"

"So, that's the fourth decimal place, 0.0006. $\cos(2º) \simeq 0.9994$?"

"Try it!" said the Mathematical Ninja.

"$0.99939$!" said the student, "so the adjacent side is 99.94m!"

The Mathematical Ninja smiled.

The student thought the Mathematical Ninja should take more holidays.


Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.


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