sin(a + ; ) = cosa, sina + sinb = 2sin±(A + B)cos±(A - B) cosa + cosb = 2cos 2 (A + B) cos (A - B) cos A - cosb = -2 sin ±(A+ B) sin i(a - B)

Transcription

1 Trigonometr Formuas. Definitions an Funamenta entities Sine: sin(}=~= - - r csc (} Cosine: Tangent:. entities sin ( -(}) = -sin(}, cos(}= - x = -- r sec(} tan (} = x = cot (} cos ( -(}) = cos(} sin (} + cos (} =, sec (} = + tan e, csc (} = + cot (} sin (} = sin (} cos (}, (} + cos (} cos -, cos (} = cos (} - sin (}. (} - cos (} sm = sin (A + B) = sina cosb + cos A sinb sin (A - B) = sina cosb - cos (A + B) = cos A cosb - cos (A - Trigonom~tric Functions cosa sinb sina sinb B) = cos A cosb + sina sinb tan (A + B) = tan A + tanb - tana tanb tan (A _ B) = tan A - tanb + tan A tanb sin(a - ; ) = -cosa, cos(a - ; ) = sina sin(a + ; ) = cosa, cos(a + ; ) = -sina sina sinb =!cos (A - B) - ±cos (A + B) cosa cosb = cos (A - B) + cos (A+ B) sinacosb = ±sin(a - B) + ±sin(a + B) sina + sinb = sin±(a + B)cos±(A - B) sina - sinb = cos±(a + B) sini(a - B) cosa + cosb = cos (A + B) cos (A - B) cos A - cosb = - sin ±(A+ B) sin i(a - B) = sinx Raian Measure Degrees Raians n T Domain: (-co, co) Range: [-!, ] = tanx Domain: (-co, co) Range: [-!, ] = secx Domain: A rea numbers except o integer mutipes of r/ Range: (-co, co) Domain: A rea numbers except o integer mutipes of r/ Range: (-co,-!] U [, co) =}_ = ) r or 80 = r raians. s )=,:;, The anges of two common trianges, in egrees an raians.,u = CSCX ~ +-~T~O;:--t--T-'--+-73T-;:~X i\ {\ Domain: x * 0, ±r, ±r,... Range: (-co, -] U [, co) Domain: x * 0, ±r, ±r,... Range: (-co, co)

2 LMTS Genera Laws f L, M, c, an k are rea numbers an Sum Rue: im f(x) = L an im g(x) = M, then x----,c im(f(x) + g(x)) = L + M Difference Rue: im(f(x) - g(x)) = L - M Prouct Rue: Constant Mutipe Rue: im(f(x) g(x)) = L M im(k f(x)) = k L Quotient Rue: irn f(x) =.b_ M -=fc- 0 x----,c g(x) M' The Sanwich Theorem f g(x) :S f(x) :S h(x) in an open interva containing c, except possib at x = c, an if Specific Formuas f P(x) = anxn + n-jxn- + + ao, then im P(x) = P(c) = ancn + an-jcn- + f P(x) an Q(x) are ponomias an Q(c) -=fc-. P(x) P(c) hm--=- Q(x) Q(c) f f(x) is continuous at x = c, then im f(x) = f(c). x----,c 0, then + ao. then imx---->c f(x) = L. im g(x) = im h(x) = L, m--.. smx _ x-o x an. - cosx = O Ji x x-o nequaities f f (x) :S g(x) in an open interva containing c, except possib at x = c, an both imits exist, then L'Hopita's Rue f f(a) = g(a) = 0, both f' an g' exist in an open interva containing a, an g' (x) -=fc- 0 on if x -=fc- a, then im f(x) :S im g(x). Continuit f g is continuous at Lan imx--+c f(x) = L, then assuming the imit on the right sie exists. im g(f(x)) = g(l).

3 DFFERENTATON RULES Genera Formuas Assume u an v are ifferentiabe functions of x. Constant: Sum: Difference: Constant Mutipe: Prouct: Quotient: Power: Chain Rue: Trigonometric Functions! (sinx) = cosx! (cosx) = -sinx x (tanx) = sec x x (cotx) = -csc x x (secx) = secxtanx x (cscx) = -cscxcotx Exponentia an Logarithmic Functions A_ nx = _! x x - (oga X) = - - x x na nverse Trigonometric Functions (. - ) - sm x = x ~ _4_ (tan- x) = - - x + x - ( cot - x ) = --- x + x Hperboic Functions! (sinhx) = coshx x (tanhx) = sech x x (cothx) = -csch x nverse Hperboic Functions A_ (cos- x) = - x ~ A_ ( sec - x) = x x~ A_ (csc- x) = - x x~ (coshx) = sinhx x x (sechx) = -sechxtanhx x ( csch x) = -csch x coth x A_ ( sinh - x) = A_ ( cash - x) = x ~x ~ A_ (tanh- x) = - - x - x A_ (coth- x) = - - x - x Parametric Equations A_ (sech- x) = - x x~ A_ ( csch- x) = - x x~ fx = J(t) an = g(t) are ifferentiabe, then /t '/t ' - -- an -- x x/ t x x/ t

4 78 METHODS OF NTEGRATON un+. un u=--+c n+ (nci=-) u --; = n u + c 3 f e u= e+ c 4 f cos u u = sin u + c 5 f sin u u = - cos u + c 6 f sec u u = tan u + c 7 f csc u u = - cot u + c 8 f sec u tan u u = sec u + c 9 f csc u cot u u = - csc u + c 0 u = Sn. - -u + C ~a - u a u =-tan - -u + c a + u a a f tan u u= - n (cos u) + c 3 f cot u u= n (sin u) + c 4 f sec u u = n (sec u + tan u) + c 5 f csc u u= -n (csc u + cot u) + c The ast four formuas are new, an compete our ist of the integras of the six trigonometric functions. Formuas an 3 can be foun b a straightforwar process: ~n sinuu f(cosu).tanuu= =- =-n(cosu)+c. f cos u cos u cos u u cotuu=. = (sin u) =n(smu)+c. smu smu.!

5 NTEGRATON RULES Genera Formuas Zero: 0 f(x) x = 0 Orer of ntegration: i 0 6 J(x)x = - f(x)x Constant Mutipes: 6 \f(x)x = k f(x)x (an number k) b -J(x) x = - b f(x) x (k = -) Sums an Differences: \J(x) ± g(x)) x = bf(x) x ± bg(x) x Aitivit: bf(x) x + i cf(x) x = \(x ) x Max-Min nequait: f max f an min f are the maximum an minimum vaues off on [a, b], then min/ (b - a) s bf(x)x s max/ (b - a). Domination: f(x) ~ g(x) on [a, b] impies bf(x) x ~ bg(x) x f(x) ~ 0 on [a, b] impies bf(x)x ~ 0 The Funamenta Theorem of Cacuus Part f f is continuous on [a, b], then F(x) = J:J(t) t is continuous on [a, b] an ifferentiabe on (a, b) an its erivative is f(x); F'(x) = x x a f(t) t = J(x ). Part f f is continuous at ever point of [ a, b] an Fis an anti erivative off on [a, b], then bf(x ) x = F(b) - F(a). Substitution in Definite ntegras b g(b) f(g(x)) g'(x) x = g(a) f(u) u ntegration b Parts b f (x)g'(x) x = f(x)g(x) ]! - b f'(x)g(x) x