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.


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.


3 comments on “The Mathematical Ninja’s Guide To Famous Triangles

  • Joshua Zucker

    Of course you only need to remember half of them, and the value of pi/2.

    I sadly have to admit that I’m like the student in the beginning, not the ninja — even when I practiced these enough (teaching trigonometry) to remember a bunch of them, I tended to remember them in degrees and then convert to radians if needed rather than the other way around. Radians would be better, because then if I forgot I could use the power series to estimate and jog my memory.

    • Colin

      😮 I’d never do such a thing! Well, maybe sometimes, when the Mathematical Ninja isn’t looking.

      Of course, you’re right about $\pi/2$, although I’m not sure if there’s a good way to take numbers away from it quickly – possibly add 0.43 and take away 2.

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