# 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 the broadsword… well, once upon a time, it would have been a point of pride to have not a spot of blood on it.

“Interpolation!” said the Mathematical Ninja, a little bit glumly. Christmas was coming, and Christmas meant having to hang out with the Ninja family rather than do Ninja maths.

“Go on…” said the student.

“Well, $\sin(54º)$ - or rather, $\sin\left(\frac{3}{10}\pi\right)$ - is a special angle. It works out to be $\frac{1}{2}\phi$, or $\frac{\sqrt{5}+1}{4}$.”

“Oh! I know about $\phi$! It’s about 1.618, isn’t it?”

The Mathematical Ninja nodded, too focussed on Brussels sprouts to make a big deal of anything.

“So $\sin(54º)$ is about 0.809. Oh! And it’s not far from $\sin(60º)$, which is…” the student paused, remembering to give the exact form first: “… $\frac {1}{2}\sqrt{3}$, or about 0.866.”

“Not bad,” said the Mathematical Ninja. Somewhere in the distance, a slowed-down version of *So Here It Is, Merry Xmas* with added sleigh bells played. The Ninja’s mood did not improve.

“And is it ok to say it’s about a sixth of the way beween them? They’re 0.057 apart, and a sixth of the way is 0.0095. Add that on and it’s 0.8185. And you’d round up because it ends in a 5?”

“Partly,” said the Ninja. “Also, because of the shape of the graph - “ (the Ninja’s teeth were gritted, only partly to avoid frostbite) “ - linear interpolation is a slight underestimate, so it makes sense to round up.”

“Neat-o,” said the student. “Fancy a mince pie?”

The Mathematical Ninja considered how best to turn the student into the bangy bit of a cracker, but didn’t even have the energy to raise an eyebrow.