University of Edinburgh Discipline of Civil and Environmental Engineering Mathematics Databook April 2011 Contents1 1. TRIGONOMETRIC FUNCTIONS............................................................................................................. 1 2. HYPERBOLIC FUNCTIONS...................................................................................................................... 1 3. GEOMETRICAL FORMULAE.................................................................................................................. 2 4. LIMITS .......................................................................................................................................................... 3 5. SERIES........................................................................................................................................................... 3 6. DIFFERENTIATION ................................................................................................................................... 6 7. PARTIAL DIFFERENTIATION................................................................................................................. 6 8. INTEGRATION ............................................................................................................................................ 7 9. NUMERICAL ANALYSIS........................................................................................................................... 8 10. MATRIX ALGEBRA.................................................................................................................................. 12 11. VECTOR PRODUCTS............................................................................................................................... 14 12. COMPLEX VARIABLES .......................................................................................................................... 15 13. LAPLACE TRANSFORMS ....................................................................................................................... 15 14. FOURIER SERIES ..................................................................................................................................... 17 15. STATISTICS ............................................................................................................................................... 18 16. ORDINARY DIFFERENTIAL EQUATIONS ......................................................................................... 26 1. Trigonometric Functions sin( A ± B ) = sin A cos B ± cos A sin B tan( A ± B ) = cos( A ± B ) = cos A cos B m sin A sin B tan A ± tan B 1 m tan A tan B sin A + sin B = 2 sin A+ B A− B cos 2 2 cos A + cos B = 2 cos A+ B A− B cos 2 2 sin A sin B = 1 cos( A − B ) − cos( A + B ) 2 sin A − sin B = 2 cos A+ B A− B sin 2 2 cos A − cos B = −2 sin A+ B A− B sin 2 2 cos A cos B = 1 cos( A + B ) + cos( A − B ) 2 11 ( AA−+ BB)) ] [sin(AA+−BB))−+sin sin A cos BB == 2 sin( sin( 2 sin 2 x = 1 1 − cos 2 x cos 2 x = 1 1 + cos 2 x sin 3 x = 1 3 sin x − sin 3 x cos 3 x = 1 3 cos x + cos 3 x 2 2 4 4 sin 2 x + cos2 x = 1 sin x = e ix − e − ix 2i 2. Hyperbolic Functions cosh x = e x + e− x 2 cos x = sinh x = e ix + e − ix 2 e x − e− x 2 cosh ix = cos x cos ix = cosh x sinh ix = i sin x sin ix = i sinh x cosh( x ± y ) = cosh x cosh y ± sinh x sinh y sinh( x ± y ) = sinh x cosh y ± cosh x sinh y cosh( x ± iy ) = cosh x cos y ± i sinh x sin y sinh( x ± iy ) = sinh x cos y ± i cosh x sin y cosh 2 x − sinh 2 x = 1 1 3. Geometrical formulae 3.1 Triangles B c a A C b ∆ = 1 bc sin A = 1 ca sin B = 1 ab sin C Area of triangle: 2 2 2 s ( s − a )( s − b )( s − c ) where 2s = a + b + c or sin A sin B sin C = = a b c Sine Rule: a 2 = b 2 + c 2 − 2bc cos A Cosine Rule: b 2 = c 2 + a 2 − 2 ca cos B c 2 = a 2 + b 2 − 2 ab cos C 3.2 Circles Circle radius r: Perimeter = 2πr Length of arc Show More
