UsefulRelations

UsefulRelations - - cos x = X n =0 (-1) n (2 n )! x 2 n =...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
Useful Mathematical Relations Trigonometric Identities: sin( x ± y ) = sin x cos y ± cos x sin y cos( x ± y ) = cos x cos y sin x sin y tan( x ± y ) = tan x ± tan y 1 tan x tan y sin 2 x + cos 2 x = 1 , cosh 2 x - sinh 2 x = 1 cos 2 x = cos 2 x - sin 2 x, sin 2 x = 2 sin x cos x cos 2 x = 1 2 (1 + cos 2 x ) , sin 2 x = 1 2 (1 - cos 2 x ) e ix = cos x + i sin x sin x = e ix - e - ix 2 i , cos x = e ix + e - ix 2 sinh x = e x - e - x 2 , cosh x = e x + e - x 2 sin( iz ) = i sinh z, cos( iz ) = cosh z sinh( iz ) = i sin z, cosh( iz ) = cos z Binomial Expansion: ( | x | < 1 , any real n ) (1 ± x ) n = 1 ± nx ± n ( n - 1) 2! x 2 ± n ( n - 1)( n - 2) 3! x 3 + ··· 1 1 + x = 1 - x + x 2 - x 3 + ··· 1 1 - x = 1 + x + x 2 + x 3 + ··· Taylor Series: f ( x ) = X n =0 f ( n ) ( a ) n ! ( x - a ) n = f ( a ) + f 0 ( a )( x - a ) + f 00 ( a ) 2! ( x - a ) 2 + f (3) ( a ) 3! ( x - a ) 3 + ··· e x = X n =0 x n n ! = 1 + x + x 2 2! + x 3 3! + ··· sin x = X n =0 ( - 1) n (2 n + 1)! x 2 n +1 = x - x 3 3! + x 5 5!
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: - cos x = X n =0 (-1) n (2 n )! x 2 n = 1-x 2 2! + x 4 4!- sinh x = X n =0 1 (2 n + 1)! x 2 n +1 = x + x 3 3! + x 5 5! + cosh x = X n =0 1 (2 n )! x 2 n = 1 + x 2 2! + x 4 4! + Leibnitzs (or Leibnizs) Rule: x Z B ( x ) A ( x ) f ( x, t ) dt = Z B ( x ) A ( x ) f x dt + f ( x, B ) dB dx-f ( x, A ) dA dx 1...
View Full Document

This note was uploaded on 09/24/2010 for the course MMAE 501 taught by Professor Kevincassel during the Spring '10 term at Illinois Tech.

Ask a homework question - tutors are online