# The Mathematical Ninja's Guide To Famous Triangles

”$\arccos\left(\frac{1}{3}\right)$”, said the student, absent-mindedly.

The Mathematical Ninja, without thinking, said “$1.23$.”

“It says $70.5$ here… but, but,” said the student, realising that the Mathematical Ninja was not going to accept answer like that, “my calculator has… somehow… got switched into degree mode, haha, let me fix it. Ah, yes, you’re quite right. $1.23$, in the *proper* units.”

The Mathematical Ninja nodded.

“How did you know that, by the way?”

“Comes up all the time,” said the Mathematical Ninja. “One of the most common triangles, after the set squares and 3-4-5.”

“Any others I ought to know about?” It had been a close shave. There was only one way to talk the Mathematical Ninja’s temper down: get him pontificating about triangles.

”$\arctan(0.5)$ comes up a lot,” said the Mathematical Ninja. “That’s $0.464$. $\arctan(2)$ is $1.11$, too - very close to $\ln(3)$.”

“For any reason?”

“No, just coincidence.”

“What about $\arcsin\left(\frac{2}{3}\right)$?”

“Good one! That’s $0.730$.”

“Close to $\frac{\pi}{4}$, of course.”

“Of course. Its cosine is $0.841$ - you just add $0.111$ to the sine.”

“I see $\arcsin(0.4)$ a lot, too.”

“$0.411$,” said the Ninja. “They dial that for information in America. And $\arccos(0.4) = 1.16$. That looks like it ought to be nice… but it isn’t.”

The student winced. He was never going to be able to remember all of these.