useful_formulas - Things You Should Know(But Likely Forgot...

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Things You Should Know (But Likely Forgot), page 1 of ( Things You Should Know (But Likely Forgot) Trigonometric identities: cos sin tan sec ## # # )) ) ) œ " œ " cos cos sin cos sin ab œ # " œ " # ) ) ) # # sin cos sin # ß ) cos cos sin cos "" œ" # ß # cos cos cos sin sin a b ab ab +„, œ + , … + , sin sin cos cos sin a b + , „ + , sin cos sin sin a b + , œ +,  +, " # cos sin sin sin a b + +,  " # cos cos cos cos a b + " # sin sin cos cos a b + +,  +, " # sin sin sin cos ˆ‰ˆ‰ + ,œ# +, +, sin sin cos sin ˆ‰ + +, +, cos cos cos cos +, +, cos cos sin sin ,œ# +, +, cos sin cos sin ) ) œœ / œ 3 / / / / 3 „3 3 3 3 ) cos sin sin cos ) ) ww œ tan sec sec sec tan ) ) ) # Weierstrass substitution: makes .Bœ .> ß ß tan cos sin B # #" > # > "> "> "> # # Hyperbolics: even, even odd, odd B B BB „B cosh sinh cos cosh sin sinh œ sech cosh sinh tanh ß ß ß // // B B B B B sinh cosh " B cosh h cosh sinh tanh sec # # Bœ"ß Bœ" B
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Things You Should Know (But Likely Forgot), page 2 of ( "Arctrig" (inverse trigonometric functions): has domain range sin sin " " ## w " "B ab c d a b ‘ B " ß " ß ß B œ 11 È # has domain range cos cos " " w " "B c d c d a b B  "ß " ß B œ 1 È # has domain range tan tan " " # # "B w " a b a b ˆ‰ _ ß _ ß ß B œ # Inverse hyperbolics: sinh ln sinh " " # w " B" a b a b Š‹ È B B"ß _ ß_ß B œ domain and range È # cosh ln cosh " " # w " B" a b È B œ B  B  " ß Ò"ß _Ñß Ò!ß _Ñ B œ domain range È # tanh ln tanh " " "" B " #" B "B w a b a b a b ß " ß"ß ß_ domain range # Useful integration facts and tricks: for a of your choice (at which the expression is '' 0ÐBÑ.B œ 0Ð>Ñ.>G 0 B 0 well-defined) integration by parts ++ ,, + , ? .@ œ ?@ @ .? ÞÞÞ k B/ .B œ B  " /  G B.B œ B B  "  G BB a b ln ln ' +,B , .B œ  +  ,B  G ln kk B " B +B + + + " " +B .B œ  G tan sin È ' " "+ B #+ +B ln ¸¸ "B " B " " B+ ÈÈ sinh cosh One variable at a time: if then 0 Bß C œ J Bß C 0 Bß C .C œ J Bß C  1 B a b ` `C ' where is an arbitrary function of . 1B It works with more variables: ' `J BßCßD .C œ J BßCßD  1 BßD Þ a b
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Things You Should Know (But Likely Forgot), page 3 of ( Determinant tricks: for matrices, 8‚8 det det Ô× Ô × ÖÙ Ö Ù Ö Ù Ö Ù Ö Ù ÕØ Õ Ø row row row row row row "" 44 88 ãã > œ> (multiplying a row by multiplies the determinant by ) >> det det ab row row row row row row row row 34 4 3 ã œ " ã
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This note was uploaded on 02/19/2010 for the course ENGINEERIN 2Z03 taught by Professor Protas during the Spring '10 term at McMaster University.

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useful_formulas - Things You Should Know(But Likely Forgot...

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