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Calculate the constant Pi to at least 6 decimal places of accuracy, without obtaining the value of the arctangent from a modern calculator, or any modern tables, using this identity: Pi=8arctan[√2 - 1] . Show the method used in the calculation. Thanks for any help.

Guest Jan 1, 2016

Best Answer 

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Well, the method is easy enough, even though it is cumbersome and laborious. We can use Taylor series to find the arctangent of sqrt(2)-1. We could even calculate the sqrt(2) by hand, using Newton's method. Since we can obtain 6-7 accurate digits by doing 3-4 iteration of this method, we should obtain 1.41421356..... - 1=.41421356. Now, we substitute this into Taylor series as follows:

Arctangent=.41421356 - (.41421356^3/3) + (.41421356^5/5) - (.41421356^7/7)........and so on. Each term will give, for this value, approx. one accurate digit of Pi, so 7 terms will suffice. With patience and some slugging, we should get:0.39269906583409 X 8=3.141592526672....... accurate to 6 decimal places.

Guest Jan 1, 2016
 #1
avatar
+10
Best Answer

Well, the method is easy enough, even though it is cumbersome and laborious. We can use Taylor series to find the arctangent of sqrt(2)-1. We could even calculate the sqrt(2) by hand, using Newton's method. Since we can obtain 6-7 accurate digits by doing 3-4 iteration of this method, we should obtain 1.41421356..... - 1=.41421356. Now, we substitute this into Taylor series as follows:

Arctangent=.41421356 - (.41421356^3/3) + (.41421356^5/5) - (.41421356^7/7)........and so on. Each term will give, for this value, approx. one accurate digit of Pi, so 7 terms will suffice. With patience and some slugging, we should get:0.39269906583409 X 8=3.141592526672....... accurate to 6 decimal places.

Guest Jan 1, 2016

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