Laplace Transforms f(t) F(s) ) ( ' t f ) 0 ( ) ( f s sF - ) ( ' ' t f ) 0 ( ' ) 0 ( ) ( 2 f sf s F s - - ) ( Impulse Unit t d 1 ) ( 1 Step Unit t s 1 t 2 1 s at e - a s 1 at te - ( 2 1 a s ) sin( t w 2 2 w w s ) cos( t w 2 2 w s s ( at e a - - 1 1 ( a s s 1 ) sin( t e at w - 2 2 ) ( w w a s ) cos( t e at w - 2 2 ) ( w a s a s t e n t n n 1 sin 1 2 2 z w z w zw - - - 2 2 2 2 n n n s s w zw w ( f z w z zw - - - - - t e n t n 1 sin 1 1 2 2 z z f 2 1 1 tan - = - 2 2 2 n n s s s w zw ( f z w z zw - - - - t e n t n 1 sin 1 1 1 2 2 z z f 2 1 1 tan - = - ( 2 2 2 2 n n n s s s w zw w Conversion Factors: lb 2.2046 kg 1 = lb 32.174 slug 1 = ft - lb 32.174 ft - slug 1 2 2 = lbf N 2248 . 0 1 = lb - ft 0.7376 m - N 1 f = s - 1W J 1 m - N 1 = = kcal 10 X 2.389 J 1 4 - = lb - ft 778 Btu 1 f = J/s 1 W 1 = s lbf ft / 550 hp 1 - = W 7 . 745 hp 1 = EULER IDENTITIES x j x e jx sin cos = x j x e jx sin cos - = - 2 cos jx jx e e x - = j e e x jx jx 2 sin - - = TAYLOR SERIES ... ! 4 ! 2 1 cos 4 2 - - = q q q ... ! 5 ! 3 sin 5 3 - - = q q q q EXPANSION FORMULAS b a b a b a sin sin cos cos ) cos( m = ± a b b a b a cos sin cos sin ) sin( ± = ± DOUBLE/HALF ANGLE a a a cos sin 2 2 sin = 1 cos 2 sin 2 1 sin cos 2 cos 2 2 2 2 - = - = - = a a a a a 2 2 cos 1 cos 2 a a = 2 2 cos 1 sin 2 a a - = LAW OF COSINES C ab b a c cos 2 2 2 2 - = OTHER ) sin( sin cos q w w w = t A t B t A o 2 2 B A A o = ( o o 90 0 tan 1 ≤ ≤ = - q q B A LOG/ NATURAL LOG x y log 10 = = x y x y ln = = x e y DECIBEL(dB) x(dB)/20 10 x x log 20 ) ( = = dB x
