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Unformatted text preview: is in general impos s ible us ing the given tools , by field theory .
A formula for c omputing the trigonometric identities for the third-angle ex is ts , but it requires finding the z eroes of the c ubic equation , where x is the value of the s ine func tion at s ome angle and d is the k nown value of the s ine func tion at the triple angle. However, the dis c riminant of this equation is negative, s o this equation has three real roots (of whic h
only one is the s olution within the c orrec t third-c irc le) but none of thes e s olutions is reduc ible to a real algebraic ex pres s ion, as they us e intermediate c omplex numbers under the c ube
roots , (whic h may be ex pres s ed in terms of real-only func tions only if us ing hy perbolic func tions ). Sine, cosine, and tangent of multiple angles  For s pec ific multiples , thes e follow from the angle addition formulas , while the general formula was given by 16th c entury Frenc h mathematic ian Vieta. In eac h of thes e two equations , the firs t parenthes iz ed term is a binomial c oeffic ient, and the final trigonometric func tion equals one or minus one or z ero s o that half the entries in eac h of the
s ums are removed. tan nθ c an be written in terms of tan θ us ing the rec urrenc e relation: c ot nθ c an be written in terms of c ot θ us ing the rec urrenc e relation: Chebyshev method  en.wikipedia.or g /wiki/List_of_tr ig onometr ic_identities 5/12 12/7/13 List of tr ig onometr ic identities - Wikipedia, the fr ee encyclopedia The Cheby s hev method is a rec urs ive algorithm for finding the nth multiple angle formula k nowing the (n − 1)th and (n − 2)th formulae.
The c os ine for nx c an be c omputed from the c os ine of (n − 1)x and (n − 2)x as follows : Similarly s in(nx) c an be c omputed from the s ines of (n − 1)x and (n − 2)x For the tangent, we have: where H/K = tan(n − 1)x. Tangent of an average  Setting either α or β to 0 gives the us ual tangent half-angle formulæ . Viète's infinite product  Power-reduction f ormula  Obtained by s olving the s ec ond and third vers ions of the c os ine double-angle formula.
S i ne Cosine Othe r and in general terms of powers of s in θ or c os θ the following is true, and c an be deduc ed us ing De Moivre's formul...
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