Today on Twitter John D. Cook, writing as @AlgebraFact, posted the following tweet:

In radians, sin(11) is very nearly -1.

(It happens to be -0.9999902…)

I thought that was awesome! So, I (@divbyzero) replied that cos(333) is approximately 1. (It is 0.999961…)

Then @michiexile chimed in, pointing out that cos(355) is closer to -1 than cos(333) is to 1. (It is -0.9999999995…)

Finally, I countered with cos(103993), which is 0.9999999998…

So what’s going on here?

All of this comes from the continued fraction for ,

.

It is well known that the convergents of a continued fraction (i.e., the fraction you obtain by truncating a continued fraction at some point) are the best rational approximations for the number. The first ten convergents of are

, , , , , , , , , .

Notice that , so . Thus . This was John D. Cook’s observation.

Similarly, , so , and . The other two approximations come from the next two convergents.

After the conversation on Twitter, I started playing a little more.

First, I noticed that

.

()

This comes from the third convergent. Since implies that , .

Actually, we can do better than that:

.

()

The approximation, implies that , so .

Next, I noticed that

.

()

This comes from , which implies that . Thus

Finally, I noticed that

.

()

In this case, gives , and . We obtain the result by substituting the previous approximation for .

Of course, we also have

.

()

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That’s great! You really ran with this.

One footnote: Good approximations to pi lead to great approximations to 1 when you take sines. Taylor series says sin(pi/2 + h) is approximately 1 – 0.5 h^2 for small h. So the error in the approximation for pi gets squared.

By:

Johnon February 17, 2010at 12:17 am

John,

Thanks for pointing that out. I did notice that some approximations were better than others, but hadn’t thought about why.

Dave

By:

Dave Richesonon February 17, 2010at 11:06 am

Reminds me of my favorite calculator trick.

Set your calculator to degrees mode (NOT radians).

Type in a bunch of 5′s: 555555, or whatever.

Press “1/x”.

Press “sin”.

Examine the mantissa of the result. Magic!

By:

Nemoon February 17, 2010at 3:27 pm

Excellent, I’d never seen that before! Good use of and the radians-to-degrees conversion. I’ll have to show that to my students.

By:

Dave Richesonon February 17, 2010at 3:38 pm

[...] math behind a neat calculator trick I received an interesting comment on yesterday’s blog post from Nemo. It was a cool calculator trick that I’d never seen before. Nemo wrote: Reminds me [...]

By:

The math behind a neat calculator trick « Division by Zeroon February 17, 2010at 9:50 pm

Found a small typo in your typesetting: the CF of pi is 3; 7, 15, 1, 292 – you forgot the 1 after the 15 in your nested fraction.

By:

Walton February 24, 2010at 9:06 am

Wow, good eyes. Thank you for catching that. I updated the post.

By:

Dave Richesonon February 24, 2010at 9:51 am

[...] with continued fractions Posted on June 5, 2010 by amca01 In an excellent blog post earlier this year, Dave Richeson commented on the [...]

By:

Approximations with continued fractions « Alasdair's musingson June 4, 2010at 8:39 pm

Awesome.. You guys really doing some good approximation and bringing some fun to it. How I wish Twitter had been there in my school days.

By:

Prasad Kulkarnion March 7, 2011at 1:51 am

Hi, seems like you have done gross approximations.

Tan 4pi/3 is no way near sec 4pi/3 . Same goes for 5pi/3. Unless i am missing something :)

By:

Ragesh G Ron February 3, 2012at 8:29 am

Thanks for catching the typos. They’re fixed now.

By:

Dave Richesonon February 5, 2012at 9:13 pm

I read through the post several times and discussed it with my friends. I am unable to understand why have you approximated tan(111) using sec(111) and tan(69447) using sec(69447).

This is especially puzzling when I consider that computing sec(4*pi/3) in radians yields -2 and tan(4/3 * pi) is 1.7320! They aren’t even close! I’m utterly puzzled.

I’d really appreciate some insight into your thought process.

On an aside, I’d much appreciate it if you deleted my previous comment. I can’t really edit my comment, so I re-posted.

By:

batbraton February 3, 2012at 8:40 am

Thanks for catching those typos.

By:

Dave Richesonon February 5, 2012at 9:14 pm

No problem. :)

By:

batbraton February 6, 2012at 7:50 am