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.
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.
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.