List_of_trigonometric_identities

Exponential definitions function inverse function 37

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Exponential definitions Function Inverse function [37] Page 22 of 25 List of trigonometric identities - Wikipedia, the free encyclopedia 10/4/2011
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Miscellaneous Dirichlet kernel The Dirichlet kernel D n ( x ) is the function occurring on both sides of the next identity: The convolution of any integrable function of period 2 π with the Dirichlet kernel coincides with the function's n th-degree Fourier approximation. The same holds for any measure or generalized function. Weierstrass substitution Main article: Weierstrass substitution If we set then [38] Page 23 of 25 List of trigonometric identities - Wikipedia, the free encyclopedia 10/4/2011
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where e ix = cos( x ) + i sin( x ), sometimes abbreviated to cis( x ). When this substitution of t for tan( x /2) is used in calculus, it follows that sin( x ) is replaced by 2 t /(1 + t 2 ), cos( x ) is replaced by (1 t 2 )/(1 + t 2 ) and the differential dx is replaced by (2 dt )/(1 + t 2 ). Thereby one converts rational functions of sin( x ) and cos( x ) to rational functions of t in order to find their antiderivatives. See also Trigonometry Proofs of trigonometric identities Uses of trigonometry Tangent half-angle formula Law of cosines Law of sines Law of tangents Half-side formula Mollweide's formula Pythagorean theorem Exact trigonometric constants (values of sine and cosine expressed in surds) Derivatives of trigonometric functions List of integrals of trigonometric functions Hyperbolic function Prosthaphaeresis Versine and haversine Exsecant Notes 1. ^ Abramowitz and Stegun, p. 73, 4.3.45 2. ^ Abramowitz and Stegun, p. 78, 4.3.147 3. ^ Abramowitz and Stegun, p. 72, 4.3.13–15 4. ^ The Elementary Identities ( ) 5. ^ Abramowitz and Stegun, p. 72, 4.3.9 6. ^ Abramowitz and Stegun, p. 72, 4.3.7–8 7. ^ Abramowitz and Stegun, p. 72, 4.3.16 8. ^ a b c Weisstein, Eric W., "Trigonometric Addition Formulas ( )" from MathWorld. 9. ^ Abramowitz and Stegun, p. 72, 4.3.17 10. ^ Abramowitz and Stegun, p. 72, 4.3.18 11. ^ Abramowitz and Stegun, p. 80, 4.4.42 12. ^ Abramowitz and Stegun, p. 80, 4.4.43 13. ^ Abramowitz and Stegun, p. 80, 4.4.36 14. ^ Bronstein, Manual (1989). "Simplification of Real Elementary Functions". Proceedings of the ACM- SIGSAM 1989 international symposium on Symbolic and algebraic computation : 211. 15. ^ a b Weisstein, Eric W., "Multiple-Angle Formulas ( - AngleFormulas.html)" from MathWorld. Page 24 of 25 List of trigonometric identities - Wikipedia, the free encyclopedia 10/4/2011
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16. ^ Abramowitz and Stegun, p. 74, 4.3.48 17. ^ Abramowitz and Stegun, p. 72, 4.3.24–26 18. ^ Weisstein, Eric W., "Double-Angle Formulas ( - AngleFormulas.html)" from MathWorld. 19. ^ Abramowitz and Stegun, p. 72, 4.3.27–28 20. ^ Abramowitz and Stegun, p. 72, 4.3.20–22 21. ^ Weisstein, Eric W., "Half-Angle Formulas ( )" from MathWorld.
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