Check Machin-like formulas: Difference between revisions

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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:
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\over4} = \arctan{1\over2} + \arctan{1\over3}</math>
: <math>\pi/4 = 2\times\arctan(1/3) + \arctan(1/7)</math>
: <math>{\pi\over4} = 2 \arctan{1\over3} + \arctan{1\over7}</math>
: <math>\pi/4 = 4\times\arctan(1/5) - \arctan(1/239)</math>
: <math>{\pi\over4} = 4 \arctan{1\over5} - \arctan{1\over239}</math>
: <math>\pi/4 = 5\times\arctan(1/7) + 2\times\arctan(3/79)</math>
: <math>{\pi\over4} = 5 \arctan{1\over7} + 2 \arctan{3\over79}</math>
: <math>\pi/4 = 5\times\arctan(29/278) + 7\times\arctan(3/79)</math>
: <math>{\pi\over4} = 5 \arctan{29\over278} + 7 \arctan{3\over79}</math>
: <math>\pi/4 = \arctan(1/2) + \arctan(1/5) + \arctan(1/8)</math>
: <math>{\pi\over4} = \arctan{1\over2} + \arctan{1\over5} + \arctan{1\over8}</math>
: <math>\pi/4 = 4\times\arctan(1/5) - \arctan(1/70) + \arctan(1/99)</math>
: <math>{\pi\over4} = 4 \arctan{1\over5} - \arctan{1\over70} + \arctan{1\over99}</math>
: <math>\pi/4 = 5\times\arctan(1/7) + 4\times\arctan(1/53) + 2\times\arctan(1/4443)</math>
: <math>{\pi\over4} = 5 \arctan{1\over7} + 4 \arctan{1\over53} + 2 \arctan{1\over4443}</math>
: <math>\pi/4 = 6\times\arctan(1/8) + 2\times\arctan(1/57) + \arctan(1/239)</math>
: <math>{\pi\over4} = 6 \arctan{1\over8} + 2 \arctan{1\over57} + \arctan{1\over239}</math>
: <math>\pi/4 = 8\times\arctan(1/10) - \arctan(1/239) - 4\times\arctan(1/515)</math>
: <math>{\pi\over4} = 8 \arctan{1\over10} - \arctan{1\over239} - 4 \arctan{1\over515}</math>
: <math>\pi/4 = 12\times\arctan(1/18) + 8\times\arctan(1/57) - 5\times\arctan(1/239)</math>
: <math>{\pi\over4} = 12 \arctan{1\over18} + 8 \arctan{1\over57} - 5 \arctan{1\over239}</math>
: <math>\pi/4 = 16\times\arctan(1/21) + 3\times\arctan(1/239) + 4\times\arctan(3/1042)</math>
: <math>{\pi\over4} = 16 \arctan{1\over21} + 3 \arctan{1\over239} + 4 \arctan{3\over1042}</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\over4} = 22 \arctan{1\over28} + 2 \arctan{1\over443} - 5 \arctan{1\over1393} - 10 \arctan{1\over11018}</math>
: <math>\pi/4 = 22\times\arctan(1/38) + 17\times\arctan(7/601) + 10\times\arctan(7/8149)</math>
: <math>{\pi\over4} = 22 \arctan{1\over38} + 17 \arctan{7\over601} + 10 \arctan{7\over8149}</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\over4} = 44 \arctan{1\over57} + 7 \arctan{1\over239} - 12 \arctan{1\over682} + 24 \arctan{1\over12943}</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>
: <math>{\pi\over4} = 88 \arctan{1\over172} + 51 \arctan{1\over239} + 32 \arctan{1\over682} + 44 \arctan{1\over5357} + 68 \arctan{1\over12943}</math>


and confirm that the following formula is incorrect by showing '''tan'''(''right hand side)'' is ''not'' 1:
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>
: <math>{\pi\over4} = 88 \arctan{1\over172} + 51 \arctan{1\over239} + 32 \arctan{1\over682} + 44 \arctan{1\over5357} + 68 \arctan{1\over12944}</math>


These identities are useful in calculating the values:
These identities are useful in calculating the values:
: <math>\tan(a + b) = (\tan(a) + \tan(b))/(1 - \tan(a)*\tan(b))</math>
: <math>\tan(a + b) = {\tan(a) + \tan(b) \over 1 - \tan(a) \tan(b)}</math>
: <math>\tan(\arctan(a/b)) = a/b</math>
: <math>\tan\left(\arctan{a\over b}\right) = {a\over b}</math>
: <math>\tan(-a) = -\tan(a)</math>
: <math>\tan(-a) = -\tan(a)</math>


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