Testwiki:Reference desk/Archives/Mathematics/2022 March 16

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March 16

Series for pi

Machin-like_formula says:

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The following equations were derived in Angle addition formula:

$\sin (a_1+b_1)=\sin a_1\cos b_1+\cos a_1\sin b_1$

$\cos (a_2+b_2)=\cos a_2\cos b_2-\sin a_2\sin b_2$

Algebraic manipulation of these equations yields the following:

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Could someone explain what the manipulation is. -- SGBailey (talk) 13:17, 16 March 2022 (UTC)

Working backwards, the sequence of steps is as follows: The arctan equation is a case of the Arctangent addition formula, which, as explained there, can be derived from the formula for $\tan (x+y)$. And that can be derived from the starting formulas via $\tan =\sin /\cos$. Is that sufficient? --Wrongfilter (talk) 13:55, 16 March 2022 (UTC)

Wow, that is a confusing presentation. The variables $a_i,b_i$ in the $\arctan$ formula do not correspond to the synonymous variables in the angle addition formulas for $\sin$ & $\cos$. And why not directly appeal to the arctangent addition formula? Then this is immediate, apart from some elementary algebraic simplification of a compound fraction.  --Lambiam 15:47, 16 March 2022 (UTC)

I have streamlined the presentation (as well as the formatting) in that section. --JBL (talk) 16:08, 16 March 2022 (UTC)

Thanks - now I understand where 5 and 239 mathemagically came from. -- SGBailey (talk) 09:29, 17 March 2022 (UTC)