“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)1x22.”

“Where does that come from?”

“Euler series,” said the Mathematical Ninja. “Alternatively, you can say sin(x)x and use the binomial expansion on (1sin2(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 7200, which squares to 4940,000 - that’s ridiculous, isn’t it? Wait, I can round that to 5040,000, which cancels to 1800. Halve it, that’s 11,600, which is… argh! about 610,000?”

The Mathematical Ninja nodded. “Keep going!”

“So, that’s the fourth decimal place, 0.0006. cos(2º)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.