useful_formulas

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

This preview shows pages 1–4. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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.

Page1 / 7

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online