The Mathematical Ninja and $\arctan(0.4)$

It took the Mathematical Ninja a little longer than normal; the student had managed to rummage around in her bag and lay a finger on the calculator before simultaneously feeling her arm pulled away by a lasso and hearing "0.3805. Or, as a one-off, since the question is asking for degrees, $\arctan(0.4)\approx 21.8º$."

The student rubbed her arm and sighed. "Go on, then."

"As everyone knows, $\tan\br{\piby 8} = \sqrt{2}-1$."

A roll of the eyes.

"OK, there's your homework. There's also the nice result that $\tan\br{\frac{a}{b} + \frac{a-b}{a+b}} = \tan \br{\piby 4}$."

"Is it?"

"There's exercise 2 for you."

The student sighed. "So, $\tan\br{\frac{2}{5} + \frac{3}{7}}=1$. I don't really see how that helps. $\frac{3}{7}$ is what, 0.42, and that's not any easier to manipulate than the other one."

"True, true. So instead, let $\theta = \arctan\br{\frac{2}{5}}$, so that $\tan\br{2\theta} = \frac{20}{21}$."

"Oh! And that is close to 1! It's only $\frac{1}{41}$ away, so $2\theta \approx \piby 4 - \frac{1}{41}$ - assuming$\frac{1}{41}$ is small enough that $\tan(x) \approx x$."

The Mathematical Ninja picked up the lasso and furled it. There would be no further use for it in this class.

"So that makes $\theta \approx \piby 8 - \frac{1}{82}$. That would be around $\frac{31}{80}-\frac{1}{82}$, a bit more than three-eighths, so you'd call it 0.38."

"Mhm."

"And in degrees..." -- the student took care to pull a slightly disgusted face -- "that's 22.5 minus a bit. What's $\frac{57.3}{82}$, roughly? Call it $\frac{19}{27}$, a bit more than two-thirds of a degree. 21.8 degrees."

A subtle nod. The Mathematical Ninja made a note to give the Mathematical Cowboy the lasso back. Colin

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.

Share

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Sign up for the Sum Comfort newsletter and get a free e-book of mathematical quotations.

No spam ever, obviously.

Where do you teach?

I teach in my home in Abbotsbury Road, Weymouth.

It's a 15-minute walk from Weymouth station, and it's on bus routes 3, 8 and X53. On-road parking is available nearby.