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Appendices

# Appendices - Appendix A MATHEMATICAL FORMULAS A.1...

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Appendix A MATHEMATICAL FORMULAS A.1 TRIGONOMETRIC IDENTITIES tan A = sec A = sin A cos A' 1 cos A' cot A = 1 esc A = tan A 1 sin A sin 2 A + cos 2 A = 1 , 1 + tan 2 A = sec 2 A 1 + cot 2 A = esc 2 A sin (A ± B) = sin A cos B ± cos A sin B cos (A ± B) = cos A cos B + sin A sin B 2 sin A sin B = cos (A - B) - cos (A + B) 2 sin A cos B = sin (A + B) + sin (A - B) 2 cos A cos B = cos (A + B) + cos (A - B) sin A + sin B = 2 sin B A -B cos . „ A + B A - B sin A - sin B = 2 cos sin A - B A + B cos A + cos B = 2 cos cos A n ^ . A + B A -B cos A - cos B = - 2 sin sin cos (A ± 90°) = +sinA sin (A ± 90°) = ± cos A tan (A ±90°) = -cot A cos (A ± 180°) = -cos A sin (A ± 180°) = -sin A 727

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728 Appendix A tan (A ± 180°) = tan A sin 2A = 2 sin A cos A cos 2A = cos 2 A - sin 2 A = 2 cos 2 A - 1 = 1 - 2 sin 2 A tan A ± B tan (A ± B) = —— tan 2A = 1 + tan A tan B 2 tan A 1 - tan 2 A sin A = e jA - e~ iA cos A = 2/ ' — " 2 e jA = cos A + y sin A (Euler's identity) TT = 3.1416 1 rad = 57.296° \.2 COMPUX VARIABLES A complex number may be represented as z = x + jy = r/l = re je = r (cos 0 + j sin where x = Re z = r cos 0, y = Im z = r sin 0 7 = l, T = -y, The complex conjugate of z = z* = x jy = r / - 0 = re je = r (cos 0 - j sin 0) (e j9 )" = e jn6 = cos «0 + j sin «0 (de Moivre's theorem) If Z\ = x, + jy x and z 2 = ^2 + i) 1 !. then z, = z 2 only if x 1 = JC 2 and j ! = y 2 . Zi± Z 2 = (xi + x 2 ) ± j(yi + y 2 ) or nr 2 /o,
APPENDIX A 729 i j y\ or Z2 Vz = VxTjy = \Tre m = Vr /fl/2 2 n = (x + /y)" = r" e ;nfl = r n /nd (n = integer) z = ( X + yj,)"" = r 1/n e ^ " = r Vn /din + 27rfc/n (t = 0, 1, 2, , n - In (re'*) = In r + In e 7 * = In r + jO + jlkir (k = integer) A3 HYPERBOLIC FUNCTIONS sinhx = tanh x = u ~ - e x - e' x 2 sinh x cosh x 1 coshx = COttlJt = e x 1 sechx = tanhx 1 coshx cosjx = coshx coshyx = cosx sinhx sinyx j sinhx, sinhyx = j sinx, sinh (x ± y) = sinh x cosh y ± cosh x sinh y cosh (x ± y) = cosh x cosh y ± sinh x sinh y sinh (x ± jy) = sinh x cos y ± j cosh x sin y cosh (x ± jy) = cosh x cos y ±j sinh x sin y sinh 2x sin 2y tanh (x ± jy) = ± / cosh 2x + cos 2y cosh 2x + cos 2y cosh 2 x - sinh 2 x = 1 sech 2 x + tanh 2 x = 1 sin (x ± yy) = sin x cosh y ± j cos x sinh y cos (x ± yy) = cos x cosh y + j sin x sinh y L

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730 Appendix A A.4 LOGARITHMIC IDENTITIES If | log xy = log x + log y X log - = log x - log y log x" = n log x log 10 x = log x (common logarithm) log e x = In x (natural logarithm) l,ln(l + x) = x A.5 EXPONENTIAL IDENTITIES e x = where e == 2.7182 X ~f" e [e In x 2 2 ! 4 V = 1" = x 3 " 3! + e x+y X x 4 4! A.6 APPROXIMATIONS FOR SMALL QUANTITIES If \x\ <Z 1, (1 ± x) n == 1 ± ra ^ = 1 + x In (1 + x) = x sinx sinx == x or hm = 1 >0 X COS 1 tanx — x
APPENDIX A «K 731 A.7 DERIVATIVES If U = U(x), V = V(x), and a = constant, dx dx dx dx dx d\U \ U dx dx V 2 ~(aU n ) = naU n ~ i dx dx U dx d 1 dU — In U = dx U dx d v .t/, dU — a = d In a dx dx dx dx dx dx — sin U = cos U dx dx d dU —-cos U = -sin U dx dx d , dU -tan U = sec £/ dx dx d dU — sinh U = cosh [/ — dx dx — cosh t/ = sinh {/ dx dx d . dU tanh[/ = sech 2 t/ — <ix dx dx

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732 Appendix A A.8 INDEFINITE INTEGRALS lfU= U(x), V = V(x), and a = constant, a dx = ax + C UdV=UV- | VdU (integration by parts) U n+l U n dU = + C, n + - 1 n + 1 dU U = In U + C a u dU = + C, a > 0, a In a e u dU = e u +C e ax dx = - e ax + C a xe ax dx = —r(ax - 1) + C x e ax dx = ( a 2 x 2 - lax + 2) + C a' In x dx = x In x x + C sin ax cfcc = — cos ax + C a cos ax ax = sin ax + C tan ax etc = - In sec ax + C = — In cos ax + C a a sec ax ax = In (sec ax + tan ax) + C a
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