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

= 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)