sin( α + π) = sin( α) O (sin( x ) cos( x )) = sin ( x ) 2sin( x )cos( x ) + cos ( x ) = sin ( x ) + cos ( x ) 2sin( x )cos( x ) = 1 2sin( x )cos( x )

G&R vwo B deel Goniometrie en beweging C. von Schwartzenberg / spiegelen in de y -as y = sin( x f ( x = sin( x f ( x = sin( x heeft dezelfde grafiek als y = sin( x. spiegelen in de y -as y = cos( x g( x = cos( x g( x = cos( x heeft dezelfde grafiek als y = cos( x. translatie (, 0 y = sin( x h( x = sin( x + h( x = sin( x + heeft dezelfde grafiek als y = cos( x. translatie (, 0 y = cos( x j ( x cos( x j ( x cos( x = + = + heeft dezelfde grafiek als y = sin( x. translatie (, 0 y = sin( x k( x = sin( x + k( x = sin( x + heeft dezelfde grafiek als y = sin( x. translatie (, 0 y = cos( x l ( x = cos( x + l ( x = cos( x + heeft dezelfde grafiek als y = cos( x. a c sin( α O α α cos( α = sin( α α + O cos( α + α cos( α cos( α + = cos( α cos( α b d sin( α α + O α sin( α + = sin( α O α cos( α sin( α sin( α + De stelling van Pythagoras geeft: sin ( α + cos ( α = a sin( x + = cos( x + = cos( x. b cos( x + = sin( x + + = sin( x +. c sin( x = sin( x + = sin( x + = cos( x + = cos( x. d cos(x + = cos(x + + = cos(x + = sin(x + + = sin(x + = sin( x +. a (sin( x cos( x = sin ( x sin( x cos( x + cos ( x = sin ( x + cos ( x sin( x cos( x = sin( x cos( x sin cos sin cos sin = + = + = tan ( x +. cos ( x cos ( x cos ( x x + x x x x b ( cos( x sin ( x + tan ( x cos ( x = + cos ( x cos ( x sin ( x. = + = cos ( x c a sin ( x + cos( x = cos ( x + cos( x. cos ( x + sin( x = sin ( x + sin( x = sin ( x + sin( x = sin ( x + sin( x. b c sin ( x + cos ( x + cos( x = cos ( x + cos ( x + cos( x = cos ( x + cos ( x + cos( x = cos ( x + cos( x. sin( x = cos( x + cos( x = cos( x + + cos( x = cos( x + hiernaast gaat het verder x = x + + k x = x + k x = + k x = + k x = + k x = + k x op [0, ] geeft x = x = x = x =.

G&R vwo B deel Goniometrie en beweging C. von Schwartzenberg / 7a sin( x + = cos( x 7d cos( x = cos(x + cos( x + = cos( x cos( x = cos(x + + x = x + + + k x = x + k x = x + k x = x + k x = + + k x = + k x = k x = k x = k x = k x = + k x = + k x op [0, ] x = 0 x = x = x op [0, ] x = + x = x = x =. x =. 7b sin( x = cos( x 7e sin( x + = sin( x cos( x = cos( x + sin( x = sin( x sin( x = x = x + + k x = x + k x = + k x = + k x = + k x = + k x = + k x = + k 8 x op [0, ] x = x =. x op [0, ] x = x = x = x = 7 x = x = 7. 8 8 8 8 7c sin ( x + cos( x = 7f sin ( x + cos ( x + cos( x = 0 cos ( x + cos( x = ( cos ( x + cos ( x + cos( x = 0 cos ( x + cos( x = 0 cos ( x + cos ( x + cos( x = 0 cos( x ( cos( x = 0 cos ( x + cos( x + = 0 cos( x = 0 cos( x = cos ( x cos( x = 0 x = + k x = + k x = ( cos( x ( cos( x + = 0 + k cos( x = (kan niet cos( x = x op [0, ] x = x = x = x =. x = + k x op [0, ] x =. 8a cos( t = sin( t 8b cos( t = cos( t t = t + k t = t + + k t = + k t = + k t = + k t = + k t op [0, ] t = t = t = t = t =. sin( t = cos( t cos( t = cos( t + t = t + + k t = t + k t = + k 7t = + k t = 9 + k t = + k 7 7 t op [0, ] t = t = t =. 7 9a sin( x = sin( x 9d --- sin( x = 0 9e sin( x = sin( x + x = k x = x + + k x = ( x + + k 9b sin( x = sin( x x = x + k x = x + k x = + k x = x + k x = k x = + k x = + k x = + k x = k x = + k x = + k x = + k 9 9c --- 9f --- 0a verm. in de y -as, verm. in de x -as, y = cos( x y = cos( x g( x = cos( x. 0b Zie de schets hiernaast. f g 0c f ( x = sin( x = x = + k x = + k x op [0, ] x = 7 = x = =. 0e f ( x = g( x sin( x = cos( x 0d g( x = cos( x = cos( x = cos( x + cos( x = x = x + + k x = x + k x = + k x = + k x = + k x = + k x = + k x = + k x = + k x = + k x op [0, ] x = x = x = x =. x op [0, ] x = x = x =. f ( x g( x (zie de schets x = x

G&R vwo B deel Goniometrie en beweging C. von Schwartzenberg / a f ( x = sin( x heeft evenwichtsstand 0; amplitude ; periode = en beginpunt (, 0. g( x = cos( x + heeft evenwichtsstand 0; amplitude ; periode en laagste punt (,. b Gebruik de plot hiernaast voor een schets van de grafieken. c f ( x = sin( x = 0 g( x = cos( x + = 0 x = k g cos( x + = 0 x = + k x + = + k f x = + k x = + k x op [0, ] x = x = x =. x op [0, ] x = x =. De nulpunten van f zijn, en. De nulpunten van g zijn en x =. d f ( x = sin( x = e f ( x = g( x sin( x = cos( x + x = + k x = + k cos( x = cos( x + + x = + k x = + k cos( x = cos( x + x = + k x = 7 + k x = x + + k x = x + k x op [0, ] x = x = x = 7. x = + k x = + k f ( x > (zie plot < x < 7 < x. x = + k x = + k 9 x op [0, ] x = 0 x = x =. 9 9 f ( x < g( x (zie plot < x <. 9 9 a AB = ya yb = ya = sin( α. b AB = OA + OB OA OB cos AOB b cos( t + u = cos( t cos( u sin( t sin( u ( sin( α = + cos( α u vervangen door u geeft sin ( α = cos( α cos( t + u = cos( t cos( u sin( t sin( u cos( α = sin ( α sin( t + u = cos( t sin( u sin( t cos( u cos( α = sin ( α. sin( t + u = cos( t sin( u + sin( t cos( u sin( t + u = sin( t cos( u + cos( t sin( u. a cos( t u = cos( t cos( u + sin( t sin( u u vervangen door u geeft cos( t ( u = cos( t cos( u + sin( t sin( u cos( t + u = cos( t cos( u + sin( t sin( u cos( t + u = cos( t cos( u sin( t sin( u. c sin( t + u = sin( t cos( u + cos( t sin( u u vervangen door u geeft sin( t + ( u = sin( t cos( u + cos( t sin( u sin( t u = sin( t cos( u + cos( t sin( u sin( t u = sin( t cos( u cos( t sin( u. a sin( t + u = sin( t cos( u + cos( t sin( u t en u beide vervangen door A geeft sin( A + A = sin( A cos( A + cos( A sin( A sin( A = sin( A cos( A. cos( t + u = cos( t cos( u sin( t sin( u t en u beide vervangen door A geeft cos( A + A = cos( A cos( A sin( A sin( A cos( A = cos ( A sin ( A. a cos( A = cos ( A cos ( A = cos( A cos ( A = + cos( A. b b cos( A = cos ( A sin ( A (zie a cos( A = cos ( A cos ( A cos( A = cos ( A + cos ( A cos( A = cos ( A. cos( A = cos ( A sin ( A cos( A = sin ( A sin ( A cos( A = sin ( A (zie ook b. cos( A = sin ( A sin ( A = cos( A sin ( A = cos( A. a sin( x cos( x = sin( x c sin( x cos( x = sin( x sin( x = sin( x sin( x = sin( x x = x + k x = x + + k x = + k x = + + k x = + k x = + + k. sin ( x = cos( x + Gebruik: cos( A = sin ( A sin ( A = cos( A cos( x = cos( x + cos( x = cos( x = x = + k x = + k.

G&R vwo B deel Goniometrie en beweging C. von Schwartzenberg / b cos ( x = cos( x + cos ( x = cos ( x + cos ( x = cos ( x = cos( x = ± = ± = ± x = + k x = + k. 8 d sin( x + cos( x = sin ( x + sin( x cos( x + cos ( x = sin ( x + cos ( x + sin( x = sin( x = x = + k x = + k x = + k x = + k. 7a Evenwichtsstand ; amplitude ; periode en beginpunt (hoogste punt bij cosinus (0,. Dus y = + cos( x. 7b cos( A = sin ( A sin ( A = cos( A y = sin ( x + cos( x = cos( x + cos( x = + cos( x. 8 sin( x = sin( x + x = sin( x cos( x + cos( x sin( x = sin( x cos( x cos( x + sin ( x sin( x = sin( x cos ( x + sin( x sin ( x = sin( x sin ( x + sin( x sin ( x = sin( x sin ( x + sin( x sin ( x = sin( x sin ( x. 9b cos( x = cos( x + x = cos( x cos( x sin( x sin( x = cos ( x cos( x sin( x cos( x sin( x = cos ( x cos( x sin ( x cos( x = cos ( x cos( x cos ( x cos( x = cos ( x cos( x cos( x + cos ( x = cos ( x cos( x. 9a cos( A = sin ( A sin ( A = cos( A. y = cos( x sin ( x = cos( x ( cos( x = cos( x. 9b staat hierboven uitgewerkt. 0a cos( A = cos( A +. 0c cos( A = sin ( A of sin ( A = cos( A. 0b sin( A = cos( A. 0d cos( A = cos ( A of cos ( A = + cos( A. 0e cos( A = sin( A +. Voor B geldt: xb = xa en yb = ya. Voor C geldt: xc = xa en yc = ya. a Voor elke p geldt: f ( p = p cos( p = p cos( p = f ( p. f ( p = f ( p f ( p + f ( p = 0 f is (puntsymmetrisch in O. b Voor elke p geldt: g( p = p sin( p = p sin( p = p sin( p = g( p g is (lijnsymmetrisch in de y -as. a b a Voor elke p geldt: f ( p = cos ( psin( p = cos ( p sin( p = cos ( psin( p = f ( p. f ( p = f ( p f ( p + f ( p = 0 f is symmetrisch in O. + p f ( p = cos ( psin( p = cos ( + psin( + p = f ( + p. Dus f is symmetrisch in de lijn x =. Gebruik: cos( p = cos( + p (kwadr. cos ( p = cos ( + p cos ( p = cos ( + p. O f ( p = sin( p cos( p = sin( cos cos( sin cos( cos + sin( sin ( p p ( p p ( p p ( p p = cos sin cos + sin = cos( p sin( p cos( p + sin( p = cos( p. p

G&R vwo B deel Goniometrie en beweging C. von Schwartzenberg / b f ( + p = sin( + p cos( + p = sin( cos + cos( sin cos( cos sin( sin ( p p ( p p ( p p ( p p = cos + sin cos sin = cos( p + sin( p cos( p sin( p = cos( p. Voor elke p geldt: f ( p = f ( + p f is symmetrisch in de lijn x =. a f ( p = cos( p + sin( p + = cos( cos( p + sin( sin( p + sin( cos( p cos( sin( p + = cos( p + sin( p + cos( p sin( p + = cos( p +. f ( + p = cos( + p + sin( + p + = cos( cos( p sin( sin( p + sin( cos( p + cos( sin( p + = cos( p sin( p + cos( p + sin( p + = cos( p +. Er geldt: f ( p = f ( + p f is symmetrisch in de lijn x =. Alternatieve uitwerking f ( + p = cos( + p + sin( + p + (gebruik de eenheidscirkel hiernaast = sin( p + cos( p + = cos( p + sin( p + = f ( p. O f ( p = f ( + p f is symmetrisch in de lijn x = b f ( p = cos( p + sin( p + = cos( cos( p + sin( sin( p + sin( cos( p cos( sin( p + = cos( p + sin( p + cos( p sin( p + = sin( p +. f ( + p = cos( + p + sin( + p + = cos( cos( p sin( sin( p + sin( cos( p + cos( sin( p + = cos( p sin( p + cos( p + sin( p + = sin( p +. f ( p + f ( + p = sin( p + sin( p + = f is symmetrisch in het punt (,. + p p a b f ( x = sin( x waarschijnlijk is f '( x = cos( x. c f ( x = cos( x waarschijnlijk is f '( x = sin( x. 7a Zie de plot van y hiernaast. 7b f ( x = sin( x f '( x = cos( x. 7c f ( x = cos( x f '( x = sin( x. 8 f ( x = cosx = sin( x + f '( x = cos( x + = sin( x. 9 f ( x = sin( ax + b f '( x = cos( ax + b a = a cos( ax + b. g( x = cos( ax + b g '( x = sin( ax + b a = a sin( ax + b. 0a f ( x = + sin( x f '( x = cos( x = 8cos( x. 0b g( x = 0 + cos( ( x g '( x = sin( ( x = 8sin( ( x. 0c h( x = x cos( x h'( x = cos( x + x sin( x = cos( x x sin( x. 0d j ( x = x cos( x j '( x = cos( x + x sin( x = cos( x x sin( x.

0e G&R vwo B deel Goniometrie en beweging C. von Schwartzenberg / k( x = x sin( x k '( x = x sin( x + x cos( x = x sin( x + x cos( x. 0f l ( x = x sin( x l '( x = sin(x + x cos(x = sin(x + x cos(x. a f ( x = tan( x f '( x = =. cos ( x cos ( x tan tan ' tan sin( x sin( x g x = x = x g x = x = = cos cos( x. x cos ( x cos ( x b sin( x c h( x = cos( x tan( x = cos( x = sin( x h'( x = cos( x. cos( x I f ( x = sin ( x = sin( x sin( x f '( x = cos( x sin( x + sin( x cos( x = sin( x cos( x. II f ( x = sin ( x = sin( x f '( x = sin( x cos( x. III f ( x = sin ( x = cos( x f '( x = sin( x = sin( x. Mijn persoonlijke voorkeur gaat uit naar II omdat in dit geval f ( x niet hoeft te worden herschreven. a f ( x = cos ( x = cos( x f '( x = cos( x sin( x = sin( x cos( x. b g( x = sin ( x = sin( x g '( x = sin( x cos( x. c h( x = + cos ( x = + cos( x h'( x = cos( x sin( x = sin( x cos( x. d j ( x = x + sin ( x = x + sin( x j '( x = + sin( x cos( x. f ( x = sin ( x = sin( x f '( x = sin ( x cos( x. a g( x = x sin ( x = x sin( x g '( x = sin ( x + x sin( x cos( x = sin ( x + x sin( x cos( x. b c h( x = cos ( x = cos( x h'( x = cos( x sin( x = sin( x cos( x. d j ( x = cos ( x = cos( x j '( x = cos( x sin( x x = x sin( x cos( x. f ( x = sin ( x + sin( x = sin( x + sin( x f '( x = sin ( x cos( x + cos( x a g( x = sin ( x cos( x = sin( x cos( x b = cos( x cos ( x + cos( x = cos( x cos ( x. g '( x = sin( x cos( x cos( x + sin ( x sin( x = sin( x cos ( x sin ( x = sin( x sin ( x sin ( x = sin( x sin ( x sin ( x = sin( x sin ( x. tan( x sin( x sin sin cos sin cos cos ' cos sin x x x x x. cos ( x c h x = = = = ( x h x = ( x x = a f ( x = + sin( x heeft evenwichtsstand ; amplitude ; periode en beginpunt (,. b Horizontale raaklijnen in de toppen bij x = + = en x = + =. f 7a f ( x = + sin( x = + sin(( x heeft evenwichtsstand ; amplitude ; periode = en beginpunt (,. Hoogste punten zijn ( + + k, + = ( + k, 0. Laagste punten zijn ( + + k, = ( + k,. De toppen zijn ( + k, 0 en ( k,.

G&R vwo B deel Goniometrie en beweging C. von Schwartzenberg 7/ 7b g( x = + cos( x + = + cos( ( x + heeft evenwichtsstand ; amplitude ; periode = en beginpunt (hoogste punt (, + = (,. Hoogste punten zijn ( + k, + = ( + k,. Laagste punten zijn ( + + k, = ( + k,. 7c h( x = sin( x + heeft evenwichtsstand ; amplitude ; periode = en beginpunt ( +, = (,. Hoogste punten zijn ( + + k, + = ( + k,. beginpunt bij een sinus-grafiekis een punt waar de Laagste punten zijn ( + + k, = ( + k,. grafiek STIJGEND door de evenwichtsstand gaat 7d j ( x = cos( x heeft evenwichtsstand ; ampl. ; periode = en beginpunt (0 +, + = (,. Hoogste punten zijn ( + k, + = ( + k,. Laagste punten zijn ( + + k, = ( k,. beginpunt van een cosinusgrafiek is een hoogste punt 8a 8b f ( x = cos( x sin( x + f '( x = sin( x cos( x. f '( x = 0 sin( x cos( x = 0 sin( x = cos( x cos( x = cos( x + x = x + + k x = x + k x = + k x = + k x = + k x = + k. Dus x A = ; x B = ; x C = ; en x D =. y A = f = cos sin + = + = A(, ; y ( cos( sin( (, B = f = + = + = + + = B ; y ( cos sin( C = f = + = + = + + = C (, en y ( cos( sin( (, D = f = + = + = + + = D. f (0 = f = cos(0 sin(0 + = 0 + = Dus f ( x = p heeft vier oplossingen (zie ook figuur. voor p <. 9a f ( x = x + cos( x f '( x = sin( x. f '( x = 0 sin( x = 0 sin( x = sin( x = x = + k x = + k x = + k x = + k. x op [0, 7] x = x = x =. 9b f '( x = sin( x = sin( x = sin( x = x = + k x = + k x op [0, 7] x = x =. f ( x = cos ( x = cos( x f '( x = cos ( x sin( x = sin( x cos ( x. 0 f '( x = 0 sin( x cos ( x = 0 sin( x = 0 cos( x = 0 x = k x = + k x op [0, ] x = 0 x = x = x = x =. De punten zijn (0, ; (, 0; (, ; (, 0 en (,. a cos( x ( sin( x sin( x cos( x cos( x sin( x + sin ( x + cos ( x sin( x + f ( x = f '( x = = =. sin( x ( sin( x ( sin( x ( sin( x cos( x f ( x = 0 = 0 (teller = 0 cos( x = 0 x = + k. Nu x op [0, ] x = x =. sin( x x = (en y = 0 S (, 0 y = x + b sin + 0 b b, dus k: y x. rc f ' + door S (, 0 = + = = + = = = = ( sin

G&R vwo B deel Goniometrie en beweging C. von Schwartzenberg 8/ b x = (en y = 0 S (, 0 y = x + b sin + 9 0 b b, dus l : y x. rc f ' + door S (, 0 = + = = = = = = ( 9 sin( sin( x + f '( x = = 0 (teller = 0 sin( x + = 0 sin( x = sin( x = x = x = (op [0, ]. sin( x cos( randmaximum (zie plot is f = = = sin( 0 cos( maximum (zie plot is f = = = = > B sin f = [, ]. cos minimum (zie plot is f = = = =. sin( a F ( x = cos( x F '( x = sin( x = sin( x = f ( x. b G ( x = sin( x G '( x = cos( x = cos( x = g( x. a f ( x = sin( x F ( x = cos( x + c = cos( x + c. b g( x = x cos( x G ( x = x sin( x + c = x sin( x + c. c h( x = sin( x + H ( x = cos( x + + c = cos( x + + c. d j ( x = cos( x J ( x = sin( x + c = sin( x + c. + cos d = + sin = + sin 0 + 0 = + = +. 0 9 9 0 a ( x x x x x b ( x sin( x dx = x + cos( x = + cos ( + cos(0 7 = + + = +. 8 8 8 f ( x = + cos( x = 0 cos( x = cos( x = x = + k x = + k x = 9 + k x = + k x = + k x = + k. Er geldt: x op [0, ] x = x =. O ( V = ( + cos( x dx = x + sin( x = + sin( + sin( V = + + = +. g( x = sin ( x = sin( x g '( x = sin ( x cos( x f ( x. Dus g( x = sin ( x is geen primitieve van f. a bc cos( A = sin ( A sin ( A = cos( A sin ( A = cos( A. f ( x = sin ( x = cos( x F ( x = x sin( x + c = x sin( x + c. 7a 7b 7c cos( A = cos ( A cos( A + = cos ( A cos( A + = cos ( A. f ( x = cos ( x = cos( x + F ( x = sin( x + x + c = sin( x + x + c. cos( A = sin ( A sin ( A = cos( A sin ( A = cos( A. g( x = sin ( x = cos( x G( x = x sin( x + c = x sin( x + c. sin( A = sin( Acos( A sin( A = sin( Acos( A. h( x = sin( x cos( x = sin( x H ( x = cos( x + c = cos( x + c.

G&R vwo B deel Goniometrie en beweging C. von Schwartzenberg 9/ 8a 8b f ( x = tan ( x = + tan ( x F ( x = tan( x x + c. g( x = x + tan ( x = x + + tan ( x G ( x = x + tan( x x + c. 9a 9b 0 sin( A = sin( Acos( A sin( A = sin( Acos( A. sin( x cos( x dx sin( x dx cos( x cos( cos(0 = = = = + = + =. 8 0 8 8 8 8 8 0 0 cos( A = sin ( A cos( A = sin ( A cos( A = sin ( A. ( sin ( x dx ( cos( x dx sin( x x = + = + = sin + 8 8 ( sin + 8 cos( A = sin ( A sin ( A = cos( A sin ( A = cos( A. I ( L = f ( x dx = sin ( x dx = cos( x dx 0 0 0 = 0 + + 7 = 7 7 8 8 =. ( x x = sin = sin 0 sin(0 = 0 0 0 =. 8 0 8 8 V f ( x = sin( x + sin( x f '( x = sin( x cos( x + cos( x. a f '( x = 0 sin( x cos( x + cos( x = 0 cos( x sin( x + = 0 cos( x = 0 sin( x = x = + k sin( x = x = x = sin( x =. x f sin ( sin( = = + = + = + 0 =. x = f = sin + sin = ( + = = 0. sin( x = f ( x = ( + = = = =. 8 8 8 8 Randextreem: f (0 = f = 0 + 0 =. Dus B [ f =, ]. 8 b f ( x = 0 sin ( x + sin( x = 0 (stel sin( x = t t + t = 0 D = b a c = = + 8 = 9 D =. t = sin( x = ± t = sin( x = t = sin( x = (met x op [0, ] x = (zoeken we niet x = x =. cos( A = sin ( A sin ( A = cos( A. ( sin ( x sin( x dx ( cos( x sin( x dx sin( x cos( x + = + = = sin( cos( sin( cos( = = + + + =. f ( x = 0 sin ( x + sin( x + = 0 sin( x + = 0 sin( x = x = x =. f (0 = 0 + 0 + = > 0 het ingesloten gebied loopt van x = tot x =. cos( A = sin ( A sin ( A = cos( A sin ( A = cos( A. ( sin ( x sin( x dx ( cos( x sin( x dx x sin( x cos( x + + = + + = ( = sin( cos( sin( cos( = = + + = + =. 8 8 8

G&R vwo B deel Goniometrie en beweging C. von Schwartzenberg 0/ a b f ( x = sin( x heeft evenwichtsstand ; amplitude ; periode = en beginpunt (,. Zie de grafiek van f ( x = sin( x hiernaast. f ( x = 0 sin( x = 0 sin( x = sin( x = x = + k x = + k x = + k x = + k. x[0, ] x = x =. (het gevraagde gebied ligt ONDER de x -as O ( V f ( x dx = = ( + sin( x dx x cos( x x cos( x = + = = cos( cos( = cos( cos( = = + + + =. c = = ( = ( + I ( L f ( x dx sin( x dx 9sin( x 9sin ( x dx Nu is: cos( A = sin ( A sin ( A = cos( A sin ( A = cos( A. ( ( = 9sin( x + 9 cos( x dx 9 9 9 7 9 = x cos( x x sin( x ( x 8cos( x sin( x + = + 7 ( 8cos( 9 sin( 7 9 ( 8cos sin ( 8 9 9 9 ( 8 = + + = + + = 9 + 9 + = 9. a y P = sin( ct en de periode is = c =. c Of voor t = is y sin( P = = sin (de sinus heeft dan precies één periode doorlopen b Formule II: x cos( P = t. rcy = x + = AMB = ( AMB is een -- driehoek AB = MB. AM = AB = MB = = = =. xa = xm AB = en ya = ym + AB = +. t = y = x + A B a t = 0 P (,. P draait linksom. t op [0, ] driekwartcirkel. De baan van P is driekwartcirkel met middelpunt M(, en straal. t = M t = 0 b x = 0 + cos( t = 0 cos( t = cos( t = ( t op [0, ] t =. t = y sin( (0,. A = + = + = + A + c rc y = x + = bij B hoort t = en bij C hoort t =. t = x cos( en sin(. B = + = + y B = + = + t = x cos( en sin(. C = + = y C = + = C t = d x = + cos( t = (intersect of cos( t = cos( t = t = + k t = + k. t op [0, ] t =, 09 t =,9. x < (zie driekwartcirkel of plot, 09 < t <,9.

G&R vwo B deel Goniometrie en beweging C. von Schwartzenberg / 7a xp = + cos( t en yp = + sin( t. (met t in seconden 7b De eerste rondgang van t = 0 tot t =. y = 0 ( x -as + sin( t = 0 (intersect t,9 t,78. y < 0 (zie plot,9 < t <,78. Dus 0,8 seconden per rondgang. 8a t = 0 P (,. P draait linksom. t op [0, ] t op [0, ]. De baan van P is driekwartcirkel met middelpunt M(, en straal. 8b y = 0 ( x -as + sin( t = 0 sin( t = sin( t = t = + k t = + k t = + k t = (deze zoeken we + k. x cos( ( A = + = + = A, 0. 8c x = + cos( t = cos( t = cos( t = t = + k t = + k t = + k t = + k. t op [0, ] t = t =. Dus x < (zie de baan bij 8a < t <. 8d y = + sin( t = (intersect t,. Dus y < (zie 8a of de plot, < t. x = A t = M t = t = y = t =0 9a y = x + sin( t = cos( t + (met 0 t < intersect geeft dan t, en y,8 x = y = 0,8 snijpunt (0,8;,8 of t,7 en y 0,8 x = y =,8 snijpunt (,8; 0,8. 9b 9c 9d x = cos( t = cos( t = t = + k t = + k. Er geldt nu: x > voor < t <. De baan van P is een cirkel met middelpunt (0, 0 en straal. Dus ligt = deel van de cirkel rechts van de lijn x =. (omtrek van een cirkel = r De lengte van het deel rechts van de lijn x = is =. P (cos( t, sin( t en Q(cos( t, sin( t. Nu de stelling van Pythagoras: PQ = ( cos( t cos( t + ( sin( t sin( t = cos ( t cos( t cos( t + cos( t + sin ( t sin( t sin( t + sin ( t = cos ( t + sin ( t + cos( t + sin ( t cos( t cos( t sin( t sin( t = cos ( t + sin ( t + cos( t + sin ( t cos( t cos( t + sin( t sin( t = + cos( t t = cos( t. Dus PQ = cos( t. PQ = cos( t = (intersect t 0,8 t, 7. PQ = cos( t > (zie plot 0,8 < t <, 7. Dus gedurende, seconde per rondgang. 0a De translatie (, 0. 0b x Q = cos( ( t en y sin(. (met t in seconden Q = t a De omlooptijd T = = = seconden. Na = seconde (voor t = bevindt P zich in (7,. Dus x cos( ( en sin( ( P = + t y. (met t in seconden P = + t b t = x cos( (, en sin( ( P = + y,8. P = + c Na = (voor seconde t = bevindt P zich voor het eerst in (,. t = 0 t =

G&R vwo B deel Goniometrie en beweging C. von Schwartzenberg / d y = x + ( t = P ( P ( 0 ( -as sin 0 (intersect t 0, 9 x = + cos ( t, snijpunt met x -as (,; 0. t,9 x = + cos ( t, 7 snijpunt met x -as (, 7; 0. a b c xp = + cos ( yp = + sin ( ( t 0 ω = = ( t in seconden. = ( t 0 ( ( 0 0 ( t in seconden. ( 0 ( 0 ( ( ( 0 0 ( ( ( xq = + cos ( t + = + cos ( t + yq = + sin ( t + = + sin ( t + xr = + cos t = + cos ( t = + cos ( t yr = + sin t = + sin ( t = + sin ( t 0 0 Dus R loopt seconde achter op P of (omdat de omlooptijd seconde is 7 0 0 ( t in seconden. (, seconde voor op P. t = 0 (, a b c Omlooptijd is = seconde. x cos cos cos P = + t xq = + t = + t y en P sin( t = + y Q = + sin ( t = + sin t ( ( ( ( ( t in seconden. Op t = 0 heeft P een fasevoorsprong van op (,. Q heeft een fasevoorsprong van op P fasevoorsprong van Q op (, is + =. xq = + cos( ( t + = + cos( ( t + ( t in seconden. y sin Q = + ( ( t + = + sin ( ( t + Op t = 0 heeft P een fasevoorsprong van op (,. Q heeft een faseachterstand van op P fasevoorsprong van Q op (, is =. xq = + cos( ( t + = + cos( ( t + 8 ( t in seconden. y sin Q = + ( ( t + = + sin ( ( t + 8 (, (, (, O (, a De fasevoorsprong van Q op P is. De faseachterstand van R op P is. c De fasevoorsprong van 7 Q op R is + = ( > 7 Q en R is = a b Omlooptijd in stand I is seconde ω = = = 0 rad/sec. Q heeft een faseachterstand van op P x 0cos ( 0 0cos ( 0 P = 0 cos(0 t xq = t = t en yp = 0 sin(0 t y 0 sin Q = ( 0 ( t = 0sin ( 0 ( t 0cos xr = ( 0 ( t + R P t y 0sin R = ( 0 ( t + ( t in seconden. heeft een fasevoorsprong van op, dus ( in seconden. v = 08 km/uur = 0 m/s = 000 cm/s. Omlooptijd bij II is 0 = sec. 000 00 Dus ω = : = 00 = 0 rad/sec. 00 xp = 0cos(0 t ( t in seconden. yp = 0sin(0 t

G&R vwo B deel Goniometrie en beweging C. von Schwartzenberg / a De diameter van rol II is de helft van rol I, dus de omlooptijd van rol II is de helft van de omlooptijd van rol I. x 0cos( P = t = 0 cos( t cos xq = + t + en y 0sin( ( t in seconden. 0sin yq sin( P = t = t = t + b Omlooptijd rol II is sec. elke seconde loopt = 0 cm papier tussen de rollen door. Dat is per uur 0 0 0 = 000 cm 097 cm m. 7a xp = + cos( t ( t in seconden. yp = + sin( t 7b t =, xp = + cos(,, en yp = + sin(,,. 7c De baan wordt in negatieve richting (met de wijzers van de klok doorlopen. (, periode is = seconde t =, t = en t =. M 7d x = 0 ( y -as + cos( t = 0 cos( t = cos( t = A B t = + k t = + k t = + k t = + k. t = y = + sin( = + = + P (0, +. (, t = y = + sin( = = Q(0,. Alternatieve uit werking: NP = = NP = = P (0, + en Q(0,. 7e cos AMB = AMB,8 (rad De lengte van de boog onder de x -as is Ans 0,. Alternatieve uitwerking: y = 0 ( x -as + sin( t = 0 (intersect t 0, 08 t 0, 9. y < 0 (zie plot 0,08 < t < 0,9. Dus ongeveer 0,8 seconden van de seconden per omwenteling onder de x -as. De lengte van de boog onder de x -as is Ans 0,. P (, N O Q 8a T = 0 ω = = rad/min. Op t = 0 zit Frits in het hoogste punt. 0 x Frits = 7 cos ( ( t + 0 = 7 cos( t + Voor Frits geldt: ( in m y Frits 7 7 sin ( ( t 0 7 7 sin( t t inuten. = + + = + + 8b Saskia heeft = faseachterstand op Frits. 8 x Saskia = 7 cos ( ( t + 0 0 8 = 7 cos ( ( t + 0 8 ( t in minuten. y Saskia = 7 + 7 sin ( ( t + 0 0 8 = 7 + 7 sin ( ( t + 0 8 8c De omtrek van het reuzenrad wordt afgelegd in 0 minuten Dus 7 meter in 0 minuten 88 m/uur de snelheid is ongeveer 0,8 km/uur. 8d y = 00 7 + 7 sin( t = 00 (intersect t, 0 t, 0 y > 00 (zie plot, 0 < t <, 0. Dus gedurende ongeveer 0, minuten seconden boven 00 meter. Alternatieve uitwerking :, cos( α = α, 08 (rad. 7, Dus gedurende α 0 0, min. sec. boven 00 meter., α 7, 7, O

G&R vwo B deel Goniometrie en beweging C. von Schwartzenberg / Diagnostische toets Da cos( x = cos( x + = cos( x + = sin( x + + = sin( x +. Db sin( x + cos( x = sin ( x + sin( x cos( x + cos ( x = sin ( x + cos ( x + sin( x cos( x = + sin( x. + cos( x sin ( x = + cos( x cos ( x = + cos( x + cos ( x = cos ( x + cos( x. Dc Da sin( x = cos( x Dc cos( t = sin( t cos( x = cos( x sin( t + = sin( t + x = x + k x = x + k t + = t + + k t + = t + k x = + k x = + k 7 t = + k 7 t = + k 0 0 x = + k x = + k 0 t = + k 0 t = + k 0 7 7 7 7 x op [0, ] x = x = x = x = 9. 0 0 0 t op [0, 0] t = t = t = 0 t =. 7 7 7 7 Db sin ( x = sin( x + sin( x = sin ( x sin( x + = cos( x cos( x + = cos( x x + = x + k x + = x + k x = + k x = + k x = + k x = + k x op [0, ] x = x = x = 7 x = x = 9 x =. Da D sin( x + = sin( x cos( x Db sin ( x + = cos( x sin( x + = sin( x sin ( x + = sin ( x x + = x + k x + = x + k sin ( x = x = + k x = + k sin ( x = x = + k x = + k. 9 sinx = ± x = + k x = + k x = + k x = + k. sin( x cos( x sin( x sin( x sin( x cos( x sin( x cos( x cos ( x cos ( x cos( x tan( x tan( x = = = = = = cos( x. cos ( x sin ( x cos ( x sin ( x cos ( x sin ( x sin( x tan ( x cos ( x cos ( x cos ( x cos( x D f ( p + f ( + p = sin( p + cos( p + sin( + p + cos( + p = sin( p + sin( p sin( p sin( p = 0 f is symmetrisch in het punt (, 0. Da f ( x = cos( x + sin( x f '( x = sin( x + cos( x. f ( x = cos ( x = cos( x f '( x = cos( x sin( x = sin( x cos ( x. Db Dc Dd cos( x ' sin( x sin( x cos( x cos( x sin ( x cos ( x f x = f x = = =. sin( x sin ( x sin ( x sin ( x f ( x = x sin( x f '( x = x sin( x + x cos( x = x sin( x + x cos( x. De sin tan( ' cos tan sin sin cos cos tan x f x = x x f x = x x + x = x x + cos. x x Df sin( x sin( x sin( x tan( x cos( x cos( x tan( x cos ( x cos ( x cos( x cos ( x f ( x = f '( x = = sin( x sin ( x sin ( x cos ( x sin( x sin( x cos( x cos( x =. sin ( x cos ( x tan( x sin( x + tan ( x tan( x cos( x sin( x + sin( x tan ( x tan( x cos( x OF... f ( x = f '( x = =. sin( x sin ( x sin ( x

G&R vwo B deel Goniometrie en beweging C. von Schwartzenberg / D7a f ( x = sin( x heeft evenwichtsstand ; amplitude ; periode = en beginpunt ( +, = (,. Hoogste punten zijn ( 7 + + k, + = ( + k,. Laagste punten zijn ( + + k, = ( 8 + k, = ( + k,. D7b f ( x = + cos( x = + cos(( x heeft 8 evenwichtsstand ; amplitude ; periode = en beginpunt (, + = (,. 8 8 Hoogste punten zijn ( + k, + = ( + k,. 8 8 Laagste punten zijn ( + + k, = ( + k,. 8 8 D8a f ( x = sin( x sin( x f '( x = cos( x cos( x. f = sin sin = 0 0 = 0 A(, 0 en rc = f ' = cos cos = = + =. y = x + b 0 = + b = b. Dus de raaklijn in A(, 0 is y = x. door A(, 0 D8b f '( x = cos( x cos( x = 0 cos( x = cos( x cos( x = cos( x x = x + k x = x + k x = k x = k x = k x = k. Dus (zie ook figuur. x en. B = x C = y ( sin( sin( B = f = = = = B(,. y 8 C = f = sin sin = = + = C (,. D8c f '( x = cos( x cos( x = cos( x cos( x = cos ( x cos( x + = 0 cos ( x cos( x = 0 cos( x ( cos( x = 0 cos( x = 0 cos( x = x = + k x = + k x = + k. x op [0, ] x = x = x = x =. f ( = sin( sin( = = = raakpunt (,. f ( = sin sin( = 0 = raakpunt (,. f ( = sin sin( = 0 = raakpunt (,. f ( = sin( sin( = = + = raakpunt (,. D9a f ( x = sin( x + F ( x = cos( x + + c = cos( x + + c. D9b g( x = x + cos( x G( x = x + sin( x + c = x + sin( x + c. h( x = x sin ( x = x + sin ( x = x + cos( x H ( x = x + sin( x x + c. D9c D9d k( x = + tan ( x = + + tan ( x K ( x = x + tan( x + c. D0a ( sin( x + cos( x dx = cos( x + sin( x = cos + sin ( cos + sin D0b sin ( x dx cos( x dx x sin( x = = = + + = + + =. cos( A = sin ( A sin ( A = cos( A sin ( A = cos( A = sin( ( sin = + =. D ( f ( x dx = cos ( x dx = ( + cos( x dx = ( x + sin( x cos( A = cos ( A + cos( A = cos ( A + cos( A = cos ( A = ( + sin ( + sin = + 0 0 =.

G&R vwo B deel Goniometrie en beweging C. von Schwartzenberg / t = A Da De baan is een driekwartcirkel met middelpunt (, 0 en straal. t op [0, ] t op [0, ] driekwartcirkel. Db x = + cos( t = cos( t = cos( t = t = + k t = + k t = + k t = + k. Dus t = en y sin( sin( (,. A = = = A Dc In MDC geldt: sin M = CD = = M =. MC CME = =. Dus de lengte boog CE is omtrek cirkel = = = 8. t = E M O t = x = C y = t = 0 D Da Db Dc De omlooptijd is seconden ω = =. cos( xp = + ( t De parametervoorstelling voor de baan van punt P is: (met t in seconden. y sin( P = + ( t Punt Q met seconde achterstand op P geeft als parametervoorstelling voor de baan van Q : xq = + cos( ( t cos( xq = + t (met t in s y sin( econden. ( sin( Q = + t y Q = + t Punt R met een fasevoorsprong van op P geeft een voorsprong van = seconde. De parametervoorstelling voor de baan van R is: xr = + cos( ( t + x cos( ( R = + t y sin( ( (met t in seconden. R = + t + y sin( ( R = + t

G&R vwo B deel Goniometrie en beweging C. von Schwartzenberg 7/ Gemengde opgaven 0. Goniometrie en beweging sin( x cos( x sin( x Ga = = tan( x. sin ( x cos( x Gb cos ( x sin ( x = cos ( x + sin ( x cos ( x sin ( x = cos( x = cos( x. sin( x sin( x cos( x sin( x Gc = = = tan( x. + cos( x cos ( x cos( x Gd cos( x y cos( y sin( x y sin( y = cos( x y + y = cos( x. Ga sin( x cos( x = Gc cos( x + = sin( x sin( x cos( x = sin( x + + = sin( x + sin( x = x + = x + + k x + = x + k x = + k x = + k geen oplossing x = + k x = + k x = + k. x = + k. Gb cos( x = sin( x Gd cos( x sin ( x = cos( x = cos( x cos( x + cos( x = x = x + k x = x + + k cos( x = cos( x = x = + k x = + k x = + k x = + k x = + k x = + k. x = + k x = + k. 8 G7a f ( x = f ( x = g( x g( x = sin( x = sin( x = cos( x cos( x = x =. x =. x =. O V = ( sin( x dx ( cos( x dx cos( x x sin( x x + = + = cos( cos( + sin( sin( = + + + + = +. 8 8 G7b I ( L = sin d cos d ( x x + x x = ( cos( x dx + ( + cos( x dx ( x sin( x ( x sin( x sin( ( sin( + ( + sin ( + sin = + + = = + + + = +. 8 8 8 8 omtrek f '( x dx g '( x dx G7c = + + + + = + cos ( x dx sin ( x dx (fnint,9. + + + f ( x = cos( x + sin( x f '( x = cos( x sin( x + cos( x = sin( x + cos( x. G8a f '( x = 0 sin( x + cos( x = 0 cos( x = sin( x cos( x = sin( x cos( x = cos( x x = x + k (geen oplossing x = x + + k x = + k x = + k. 8 f ( x = cos ( x + sin( x = cos ( x + + sin( x = cos( x + + sin( x. absoluut maximum (zie fig G. : f ( = cos( + + sin( = + + = + 8 absoluut minimum (zie fig G. : f ( B f = [, + ]. = cos + + sin = + = 8

G&R vwo B deel Goniometrie en beweging C. von Schwartzenberg 8/ G8b f ( x = g( x cos ( x + sin( x = sin( x O ( V = ( sin( x ( cos ( x + sin( x dx cos ( x = sin( x cos ( x = sin( x cos( x = cos ( x sin( x cos( x = 0 ( sin( x cos ( x dx cos( x ( cos( x sin( x = 0 cos( x = 0 cos( x = sin( x = ( sin( x cos( x dx x = + k cos( x = cos( x x = + k x = x + k x = x + + k = cos( x sin( x x x = + k geen oplossing x = + + k x = + k x = = cos sin + cos( + sin( + + k. x op [0, ] x = x = = 0 + 0 + + =.. G8c C het midden van AB als g( p = f ( p sin( p = cos ( p + sin( p sin( p = cos ( p sin( pcos( p = cos( pcos( p sin( p cos( p = 0(voldoet niet sin( p = cos( p = tan( p = tan( p =. cos( p G9a De grafiek van f (dezelfde als in G8 is vermoedelijk lijnsymmetrisch in de verticale lijn door de eerste top rechts van de y -as. Vermoedelijk lijnsymmetrisch in de lijn x =. (zie de berekening in G8a 8 f ( + p = cos ( + p + sin( + p = cos ( + p + + sin( + p = cos( + p + sin( + p + 8 8 8 = cos( cos( p sin( sin( p + sin( cos( p + cos( sin( p + = cos( p sin( p + cos( p + sin(p + = cos( p +. f ( p = cos ( p + sin( p = cos ( p + + sin( p = cos( p + sin( p + 8 8 8 = cos( cos( p + sin( sin( p + sin( cos( p cos( sin( p + = cos( p + sin( p + cos( p sin( p + = cos( p +. f ( + p = f ( p (voor elke p de grafiek van f is symmetrisch in de lijn x =. 8 8 8 G9b Vermoedelijk puntsymmetrisch in A(,. ( A precies midden tussen de toppen bij x = en x = zie G8a 8 8 8 f ( + p = cos ( + p + sin( + p = cos( + p + sin( + p + 8 8 = cos( cos( p sin( sin( p + sin( cos( p + cos( sin( p + = cos( p + sin( p cos( p + sin( p + = sin( p +. f ( p = cos ( p + sin( p = cos( p + sin( p + 8 8 = cos( cos( p + sin( sin( p + sin( cos( p cos( sin( p + = cos( p sin( p cos( p sin( p + = sin( p +. f ( + p + f ( p = sin( p + + sin( p + = f is symmetrisch in A(,. 8 8 8 f ( x = 0 sin ( x + sin( x = 0 sin( x sin( x + = 0 sin( x = 0 sin( x = x = k x = + k x = + k. x op [0, ] nulp.: x = 0 x = x = x = x =. O ( V = sin ( x sin( x dx cos( x sin( x dx x sin( x cos( x + + + = 0 0 0 = sin cos ( 0 sin(0 cos(0 = 0 + ( 0 0 = +. G0a G0b G0c f ( x = sin ( x + sin( x = sin( x + sin( x f '( x = sin( x cos( x + cos( x f '( x = 0 sin( x cos( x + cos( x = 0 cos( x = 0 sin( x + = 0 x = + k sin( x =. x = f ( x = f = sin + sin = ( + = =. sin( x = f ( x = ( + = = =. 8 f ( x = p heeft precies vier oplossingen (zie figuur G. en de berekening hierboven voor < p < 0 0 < p <. 8

G&R vwo B deel Goniometrie en beweging C. von Schwartzenberg 9/ L(grafiek van f = + f '( x dx = + sin( x cos( x + cos( x dx (fnint,07. G0d 0 0 Ga g( x = sin( x + cos( x g '( x = cos( x sin( x. g '( x = 0 cos( x = sin( x cos( x = cos( x x = x + k x = x + + k A x = + k x = + k x = + k x = + k. x op [0, ] x = x = x = x =. Dit geeft toppen: A(,, B(,, C (, en D(,. f = cos sin = = = = A en C liggen op de grafiek van f. f cos sin = = = = Gb = ( + ( = ( + + O ( V sin( x cos( x cos ( x sin( x dx sin( x cos( x cos ( x sin( x dx ( sin( x cos( x ( cos( x dx ( sin( x cos( x dx cos( x sin( x x = + + = = = cos( sin( cos( sin( = ( = + + + + =. omtrek( V f '( x dx g '( x dx Gc = + + + = + 8cos( x sin( x cos( x dx + + cos( x sin( x dx (fnint, 7. f ( x = sin ( x + p cos( x = sin ( x + p sin ( x = p (de andere termen vallen weg voor p =. Ga p Gb fp ( x = sin ( x + p cos( x = ( sin( x + p cos( x fp '( x = sin( x cos( x p sin( x = sin( x p sin( x = p sin( x. fp '( x = ( p sin( x < p < < p < 0 > p > 0 0 < p <. a a a a f ( x dx sin ( x p cos( x dx cos( x p cos( x d x ( p cos( x dx 0 0 0 0 a = x + ( p sin( x = a + ( p sin( a ( 0 + ( p sin(0 = a + ( p sin( a. 0 Onafhankelijk van p als sin( a = 0 a = k a = k. Gegeven: a op [0, ] a = 0 a = a =. Gc p = ( + = ( + = ( + cos( x ( + sin( x sin( x cos( x cos( x sin( x sin ( x cos ( x sin( x Ga f ( x = f '( x = = =. + sin( x ( + sin( x ( + sin( x ( + sin( x f '( x = 0 (teller = 0 sin( x = 0 sin( x = x = + k x = + k. Gegeven: x op [, ] x = x =. cos( minimum (zie plot : f ( = = = = + sin( cos( B f = [, ]. maximum (zie plot : f ( = = = = + sin( 9 cos( x cos( x 9 cos( x cos( x 9 cos ( x Gb f ( x f ( x = = = = 7 + sin( x + sin( x 7 + sin( x sin( x 7 sin ( x 7 7 cos ( x = ( cos ( x cos ( x = cos ( x = cos( x = ± = ± = ± x = + k. Gegeven: x op [, ] x = x = x = x =. B C V f g D

G&R vwo B deel Goniometrie en beweging C. von Schwartzenberg 0/ Ga t op [0, ] t op [0, ]. De baan is driekwartcirkel met middelpunt (, en straal. t = Gb y = x + + sin( t = cos( t + B sin( t = cos( t cos( t = cos( t + y = x + t = t + + k t = t + k t = t = (, geen oplossing t = + k t = + k. 8 t op [0, ] t = t = A. 8 8 t = 0 t = x cos( en sin( (,. 8 P = + = + = + y P = + = + = A + t = x cos( en sin( (,. 8 P = + = + = y P = + = + = + B + Gc x = 0 ( y -as y = 0 ( x -as + cos( t = 0 + sin( t = 0 cos( t = sin( t = cos( t = sin( t = t = + k t = + k t = + k t = + k t = 7 + k t = + k t = + k t = + k t = 7 + k t = + k t = + k t = + k x > 0 (zie de baan van P 0 t < 7 y > 0 (zie de baan van P < t Uit bovenstaande regel volgt dan: x > 0 en tevens y > 0 (zie de baan van P < t < 7. 0, 0, Ga v = 0, m/s per seconde gedeelte van de cirkel rad/s = rad/s.,, Gb x =,cos( t + t = 0 ( t in seconden en x en y in meters. y =,sin( t + (,;0 Gc De volgende koker loopt cirkel = = radialen achter. x =,cos( t + ( t in seconden en x en y in meters. y =,sin( t + Gd y = 0, 8,sin( t = 0, 8 (intersect t 8, 70 t, 0. y < 0, 8 (zie plot 8, 70 < t <, 0. Dus gedurende, seconde. O wateroppervlak y = 0,8 Ga f( x = + sin ( x + cos( x = + cos( x + cos( x = + cos( x = + sin( x + = + sin(( x +. Dit geeft a =, b =, c = en d =. Gb f n = + sin + cos( n = + ( + cos( n = cos( n = n = + k n = + k. 0 < n < 0 n = n = 8 n = 0 n =. Gc f( x = + sin ( x + cos( x = + cos( x + cos( x = cos( x + cos( x. Gd O ( V = ( f( x dx 0 = ( ( cos( x + cos( x dx 0 = ( + cos( x cos( x dx 0 V = x + sin( x sin( x 0 = + sin sin(8 0 + sin(0 sin(0 = + 0 0 ( 0 + 0 0 =. O (rechthoek ABCD = = 0. Dus de grafiek van f verdeelt de rechthoek in twee gebieden met dezelfde oppervlakte.

G&R vwo B deel Goniometrie en beweging C. von Schwartzenberg / G 7a x = sin( t en y = cos( t geeft de ciirkel met middelpunt (0, 0 en straal. x = sin( t en y = cos( t geeft de ciirkel met middelpunt (0, 0 en straal. t =, de grote wijzer heeft rondgang gemaakt het is 8 minuten over. 0 G7b Wijzers (niet de eindpunten vallen over elkaar sin( t = sin( t en cos( t = cos( t. ( t = t + k t = t + k en tevens ( t = t + k t = t + k ( t = k t = + k en tevens ( t = k t = k ( t = k t = + k en tevens ( t = k t = k ( t = k t = + k en tevens ( t = k t = k t = k het eerste tijdstip na t = 0 is dus t =. G7c afstand = ( sin( t sin( t + ( cos( t cos( t = 9 sin ( t sin( t sin( t + sin ( t + 9cos ( t cos( t cos( t + cos ( t = 9( sin ( t + cos ( t + ( sin ( t + cos ( t cos( cos( ( t t + sin( t sin( t = 9 + cos( t t = cos( t. G7d Een gelijkbenige driehoek als afstand = afstand = cos( t = cos( t = cos( t = 9 cos( t = cos( t = cos( t = 9 cos( t = cos( t = Het eerste moment na t = 0 (cos(0 = volgt uit cos( t = t 0, 7 t 0,. G8a f ( x = sin( x T = (, en A(, 0. g(0 = 0 (0 = 0 de grafiek van g gaat door O. g = ( = = = de grafiek van g gaat door T. g = ( = 0 = 0 de grafiek van g gaat door A. G8b f ( x = sin( x f '( x = cos( x. g( x = x ( x = x + x g '( x = 8 x +. f '(0 = cos(0 = en g '(0 = 8 0 + = > g '(0 > f '(0. G8c ( g( x f ( x d x ( ax ( x sin( x dx ( ax ax sin( x dx ax ax cos( x = = + 0 0 0 0 = a a + cos ( a 0 a 0 + cos(0 = a a ( 0 0 + = a. ( g( x f ( x dx = 0 a = a = a =. 0