Wikipedia: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:

$\arctan {\frac {a_{1}}{b_{1}}}+\arctan {\frac {a_{2}}{b_{2}}}=\arctan {\frac {a_{1}b_{2}+a_{2}b_{1}}{b_{1}b_{2}-a_{1}a_{2}}},$ (3)

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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)