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Check Machin-like formulas: Difference between revisions

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Check Machin-like formulas (view source)

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Verify the following Machin-like formulas are correct by calculating the value of '''tan'''(''right hand side)'' for each equation using exact arithmetic and showing they equal 1:
 
: <math>\pi/4 = \arctan(1/2) + \arctan(1/3)</math>
: <math>\pi/4 = 2*\times\arctan(1/3) + \arctan(1/7)</math>
: <math>\pi/4 = 4*\times\arctan(1/5) - \arctan(1/239)</math>
: <math>\pi/4 = 5*\times\arctan(1/7) + 2*\times\arctan(3/79)</math>
: <math>\pi/4 = 5*\times\arctan(29/278) + 7*\times\arctan(3/79)</math>
: <math>\pi/4 = \arctan(1/2) + \arctan(1/5) + \arctan(1/8)</math>
: <math>\pi/4 = 4*\times\arctan(1/5) - \arctan(1/70) + \arctan(1/99)</math>
: <math>\pi/4 = 5*\times\arctan(1/7) + 4*\times\arctan(1/53) + 2*\times\arctan(1/4443)</math>
: <math>\pi/4 = 6*\times\arctan(1/8) + 2*\times\arctan(1/57) + \arctan(1/239)</math>
: <math>\pi/4 = 8*\times\arctan(1/10) - \arctan(1/239) - 4*\times\arctan(1/515)</math>
: <math>\pi/4 = 12*\times\arctan(1/18) + 8*\times\arctan(1/57) - 5*\times\arctan(1/239)</math>
: <math>\pi/4 = 16*\times\arctan(1/21) + 3*\times\arctan(1/239) + 4*\times\arctan(3/1042)</math>
: <math>\pi/4 = 22*\times\arctan(1/28) + 2*\times\arctan(1/443) - 5*\times\arctan(1/1393) - 10*\times\arctan(1/11018)</math>
: <math>\pi/4 = 22*\times\arctan(1/38) + 17*\times\arctan(7/601) + 10*\times\arctan(7/8149)</math>
: <math>\pi/4 = 44*\times\arctan(1/57) + 7*\times\arctan(1/239) - 12*\times\arctan(1/682) + 24*\times\arctan(1/12943)</math>
: <math>\pi/4 = 88*\times\arctan(1/172) + 51*\times\arctan(1/239) + 32*\times\arctan(1/682) + 44*\times\arctan(1/5357) + 68*\times\arctan(1/12943)</math>
 
and confirm that the following formula is incorrect by showing '''tan'''(''right hand side)'' is ''not'' 1:
 
: <math>\pi/4 = 88*\times\arctan(1/172) + 51*\times\arctan(1/239) + 32*\times\arctan(1/682) + 44*\times\arctan(1/5357) + 68*\times\arctan(1/12944)</math>
 
These identities are useful in calculating the values:
: <math>\tan(a + b) = (\tan(a) + \tan(b))/(1 - \tan(a)*\tan(b))</math>
: <math>\tan(\arctan(a/b)) = a/b</math>
: <math>\tan(-a) = -\tan(a)</math>
 
;Note:
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