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 Sine gives pi? (Posted on 2013-04-02)
Why does sin(1/5555555°) have a decimal that so closely resembles pi?

 See The Solution Submitted by Jer Rating: 2.0000 (1 votes)

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 Solution Comment 2 of 2 |

As x gets closer and closer to zero, sin(x) approaches x in radians.

To convert from degrees to radians, one must multiply by pi/180.

Since .1111... represents 1/9, .555... represents 5/9.

Disregarding any factor of a power of ten, 1/5555... represents 1/(5/9), or 9/5, so as more 5's are strung together in the denominator, the closer we get to 9/5 degree divided by a larger and larger power of ten. (The 9/5 itself is a limit, or asymptote.)

So in the conversion to radians we get (9/5) * pi/180 = pi/100, or just pi divided by a power of ten, to be combined with the other powers of 10, so the leading non-zero digits approach those of pi.

Here are the decimal approximations of 1/5...5, with n 5's:

n                        approx.
1                      0.20000000000000
2                      0.01818181818182
3                      0.00180180180180
4                      0.00018001800180
5                      0.00001800018000
6                      0.00000180000180
7                      0.00000018000002
8                      0.00000001800000
9                      0.00000000180000
10                      0.00000000018000

so of course when multiplied by pi/180, the digits of pi show up.

 Posted by Charlie on 2013-04-02 11:08:08

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