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Created by T. MadasCOMPLEX NUMBERSPRACTICE(part 2)Created by T. Madas
Created by T. MadasROOTSOFCOMPLEXNUMBERSCreated by T. Madas
Created by T. MadasQuestion 1z 4 16 , z » .a) Solve the above equation, giving the answers in the form a bi , where a and bare real numbers.b) Plot the roots of the equation as points in an Argand diagram.z 2 ( 1 i )Created by T. Madas
Created by T. MadasQuestion 2z5 i , z » .a) Solve the equation, giving the roots in the form r eiθ , r 0, π θ π .b) Plot the roots of the equation as points in an Argand diagram.iπz e 10 ,iπz e 2,z eCreated by T. Madasπi 910, z eπ i 310, z eπ i 710
Created by T. MadasQuestion 3z 4 4i .a) Find the fifth roots of z .Give the answers in the form r eiθ , r 0, π θ π .b) Plot the roots as points in an Argand diagram.2eπi 20, 2eπi 920Created by T. Madas, 2eπi1720, 2eπ i 720, 2e i 34π
Created by T. MadasQuestion 4z 4 4 3i .a) Find the cube roots of z .Give the answers in polar form r ( cosθ i sin θ ) , r 0, π θ π .b) Plot the roots as points in an Argand diagram.()()(z 2 cos π i sin π , z 2 cos 5π isin 5π , z 2 cos 7π i sin 7π999999Created by T. Madas)
Created by T. MadasQuestion 5The following complex number relationships are givenz4 w .w 2 2 3i ,a) Express w in the form r ( cos θ isin θ ) , where r 0 and π θ π .b) Find the possible values of z , giving the answers in the form x iy , where xand y are real numbers. 2πw 2 cos 3z 12()()6 i 2 , z 1 2 i 6 , z 122() 2π i sin 3( , 2 i 6 , z 1 6 i 22)Question 6Find the cube roots of the imaginary unit i , giving the answers in the form a bi , wherea and b are real numbers.z1 3 1 i, z2 3 1 i, z3 i2 22 2Created by T. Madas
Created by T. MadasQuestion 7Find the cube roots of the complex number 8i , giving the answers in the form a bi ,where a and b are real numbers.z1 3 i, z2 3 i, z3 2iQuestion 8z 4 8 8 3 i , z » .Solve the above equation, giving the answers in the form a bi , where a and b are realnumbers.z 3 i , z 1 3 i , z 3 i , z 1 3 iCreated by T. Madas
Created by T. MadasQuestion 9(3)z2 1 i 3 , z » .Solve the above equation, giving the answers in the form a bi , where a and b are realnumbers.z i2 2Question 10z 3 32 32 3 i , z » .a) Solve the above equation.Give the answers in exponential form z r eiθ , r 0, π θ π .b) Show that these roots satisfy the equationw9 218 0 .iπz 4 e 9 , 4eCreated by T. Madasi 7π9, 4e i 5π9
Created by T. MadasQuestion 11z7 1 0 , z » .One of the roots of the above equation is denoted by ω , where 0 arg ω a) Find ω in the form ω r eiθ , r 0, 0 θ π3π3.b) Show clearly that1 ω ω 2 ω3 ω 4 ω5 ω6 0 .c) Show further that( )ω 2 ω 5 2cos 47π .d) Hence, using the results from the previous parts deduce that1cos 2π cos 4π cos 6π .7772( )( )( )ω eCreated by T. Madasi 2π7
Created by T. MadasQuestion 12(z3 1 i 38) (1 i )5 , z » .Find the three roots of the above equation, giving the answers in the form k 2 eiθ ,where π θ π , k » .z 8 2 eiθ , θ 31π , 7π , 17π3636 36Created by T. Madas
Created by T. MadasTRIGONOMETRICIDENTITIESQUESTIONSCreated by T. Madas
Created by T. MadasQuestion 1If z cosθ isin θ , show clearly that a) z n 1zn 2 cos nθ .b) 16 cos5 θ cos 5θ 5cos3θ 10cos θ .proofCreated by T. Madas
Created by T. MadasQuestion 2It is given thatsin 5θ 16sin 5 θ 20sin 3 θ 5sin θ .a) Use de Moivre’s theorem to prove the validity of the above trigonometricidentity.It is further given thatsin 3θ 3sin θ 4sin 3 θ .b) Solve the equationsin 5θ 5sin 3θ for 0 θ π ,giving the solutions correct to 3 decimal places.θ 0, 1.095c , 2.046cCreated by T. Madas
Created by T. MadasQuestion 3The complex number z is given byz eiθ , π θ π .a) Show clearly thatzn 1zn 2 cos nθ .b) Hence show further that113cos 4 θ cos 4θ cos 2θ .828c) Solve the equation2 cos 4θ 8cos 2θ 5 0 , 0 θ 2π .θ Created by T. Madasπ 2π 4π 5π,,,3 3 3 3
Created by T. MadasQuestion 4The complex number z is given byz eiθ , π θ π .a) Show clearly thatzn 1zn 2 cos nθ .b) Hence show further that16 cos5 θ cos 5θ 5cos3θ 10cos θ .c) Use the results of part (a) and (b) to solve the equationcos 5θ 5cos 3θ 6cos θ 0 , 0 θ π .θ Created by T. Madasπ π 3π, ,4 2 4
Created by T. MadasQuestion 5De Moivre’s theorem asserts that( cosθ i sin θ )n cos nθ isin nθ , θ » ,n » .a) Use the theorem to prove the validity of the following trigonometric identity.cos 6θ 32cos6 θ 48cos 4 θ 18cos 2 θ 1 .b) Use the result of part (a) to find, in exact form, the largest positive root of theequation64 x 6 96 x 4 36 x 2 1 0 . π x cos 9 Created by T. Madas
Created by T. MadasQuestion 6Euler’s identity stateseiθ cosθ isin θ , θ » .a) Use the identity to show thateinθ e inθ 2cos nθ .b) Hence show further that32cos 6 θ cos 6θ 6cos 4θ 15cos 2θ 10 . π c) Use the fact that cos θ sin θ to find a similar expression for 32sin 6 θ . 2 d) Determine the exact value of π4sin 6 θ cos 6 θ dθ .032sin 6 θ cos 6θ 6cos 4θ 15cos 2θ 10 ,Created by T. Madas5π32
Created by T. MadasQuestion 7De Moivre’s theorem asserts that( cosθ i sin θ )n cos nθ isin nθ , θ » ,n » .a) Use the theorem to prove validity of the following trigonometric identity()sin 5θ sin θ 16cos 4 θ 12cos 2 θ 1 .b) Hence, or otherwise, solve the equationsin 5θ 10cos θ sin 2θ 11sin θ , 0 θ π .θ Created by T. Madasπ 3π,4 4
Created by T. MadasQuestion 8It is given that()sin 5θ sin θ 16cos 4 θ 12cos 2 θ 1 .a) Use de Moivre’s theorem to prove the validity of the above trigonometricidentity.Consider the general solution of the trigonometric equationsin 5θ 0 .b) Find exact simplified expressions for π 2πcos 2 and cos 2 5 5 , fully justifying each step in the workings. π 3 5 2π 3 5cos 2 , cos 2 88 5 5 Created by T. Madas
Created by T. MadasQuestion 94By considering the binomial expansion of ( cosθ i sin θ ) show thattan 4θ 4 tan θ 4 tan 3 θ1 6 tan 2 θ tan 4 θ.proofCreated by T. Madas
Created by T. MadasQuestion 10By using de Moivre’s theorem followed by a suitable trigonometric identity, showclearly that a) cos 3θ 4 cos3 θ 3cos θ .()()b) cos 6θ 2cos 2 θ 1 16cos 4 θ 16cos 2 θ 1Consider the solutions of the equation.cos 6θ 0 , 0 θ π .c) By fully justifying each step in the workings, find the exact value ofcos π cos 5π cos 7π cos 11π .12121212116Created by T. Madas
Created by T. MadasCOMPLEX LOCICreated by T. Madas
Created by T. MadasQuestion 1By finding a suitable Cartesian locus in the complex z plane, shade the region R thatsatisfies the inequalityz 3 z 3i .x y 0Created by T. Madas
Created by T. MadasQuestion 2z 1 i 4 , z » .a) Sketch the locus of the points that satisfy the above equation in a standardArgand diagram.b) Find the minimum and maximum values of z for points that lie on this locus.zmin 4 2 , zmin 4 2Created by T. Madas
Created by T. MadasQuestion 3The complex number z represents the point P ( x, y ) in the Argand diagram.Given thatz 1 2 z 2 ,show that the locus of P is given by( x 3)2 y 2 4 .proofQuestion 4The complex number z x iy represents the point P in the complex plane.Given thatz 1, z 0zdetermine a Cartesian equation for the locus of P .x2 y 2 1Created by T. Madas
Created by T. MadasQuestion 5Sketch, on the same Argand diagram, the locus of the points satisfying each of thefollowing equations.a)z 3 i 3.b) z z 2i .Give in each case a Cartesian equation for the locus.c) Shade in the sketch the region that is satisfied by both these inequalitiesz 3 i 3z z 2i( x 3)2 ( y 1)2 9Created by T. Madas, y 1
Created by T. MadasQuestion 6a) Sketch on the same Argand diagram the locus of the points satisfying each of thefollowing equations.i.z i z 2 .ii. arg ( z 2 ) π2.b) Shade in the sketch the region that is satisfied by both these inequalitiesz i z 2andπ2 arg ( z 2 ) 2π.3sketchCreated by T. Madas
Created by T. MadasQuestion 7The complex number z represents the point P ( x, y ) in the Argand diagram.Given thatz 1 2 z i ,show that the locus of P is a circle, stating its centre and radius.( x 1)2 ( y 2 )2 4 , ( 1, 2 ) , r 2Question 8z 2i 1 , z » .a) In the Argand diagram, sketch the locus of the points that satisfy the aboveequation.b) Find the minimum value and the maximum value of z , and the minimum valueand the maximum of arg z , for points that lie on this locus.z min 1 , z max 3 , arg zmin Created by T. Madasπ3, arg zmax 2π3
Created by T. MadasQuestion 9The complex number z represents the point P ( x, y ) in the Argand diagram.Given thatz 1 2 z 2i ,show that the locus of P is a circle and state its radius and the coordinates of its centre.( 13 , 83 ) ,Created by T. Madasr 2 53
Created by T. MadasQuestion 10The complex number z x iy satisfies the relationship2 z 2 3i 3 .a) Shade accurately in an Argand diagram the region represented by the aboverelationship.b) Determine algebraically whether the point that represents the number 4 i liesinside or outside this region.inside the regionCreated by T. Madas
Created by T. MadasQuestion 11Two sets of loci in the Argand diagram are given by the following equationsz z 2andz 2, z » .a) Sketch both these loci in the same Argand diagram.The points P and Q in the Argand diagram satisfy both loci equations.b) Write the complex numbers represented by P and Q , in the form a ib , wherea and b are real numbers.c) Find a quadratic equation with real coefficients, whose solutions are the complexnumbers represented by the points P and Q .z 1 3 , z 2 2 z 4 0Created by T. Madas
Created by T. MadasQuestion 12a) Sketch in the same Argand diagram the locus of the points satisfying each of thefollowing equationsi.z 3 2i 2 .ii.z 3 2i z 1 2i .b) Show by a geometric calculation that no points lie on both loci.proofCreated by T. Madas
Created by T. MadasQuestion 13The point A represents the complex number on the z plane such thatz 6i 2 z 3 ,and the point B represents the complex number on the z plane such thatarg ( z 6 ) 3π.4a) Show that the locus of A as z varies is a circle, stating its radius and thecoordinates of its centre.b) Sketch, on the same z plane, the locus of A and B as z varies.c) Find the complex number z , so that the point A coincides with the point B .() (C ( 4, 2 ) , r 20 , z 4 10 i 2 10Created by T. Madas)
Created by T. MadasQuestion 14z 2 i 5.arg ( z 2 ) 3π.4a) Sketch each of the above complex loci in the same Argand diagram.b) Determine, in the form x iy , the complex number z0 represented by theintersection of the two loci of part (a).z0 2 4iCreated by T. Madas
Created by T. MadasQuestion 15The locus of the point z in the Argand diagram, satisfy the equationz 2 i 3 .a) Sketch the locus represented by the above equation.The half line L with equationy mx 1,x 0, m 0 ,touches the locus described in part (a) at the point P .b) Find the value of m .c) Write the equation of L , in the formarg ( z z0 ) θ , z0 » , π θ π .d) Find the complex number w , represented by the point P .m 3 , arg ( z i ) Created by T. Madasπ3, w 1 3 i 1 2 2
Created by T. MadasQuestion 16The complex numbers z1 and z2 are given byz1 1 i 3andz2 iz1 .a) Label accurately the points representing z1 and z2 , in an Argand diagram.b) On the same Argand diagram, sketch the locus of the points z satisfying i. z z1 z z2 .ii. arg ( z z1 ) arg z2 .c) Determine, in the form x iy , the complex number z3 represented by theintersection of the two loci of part (b).() (z3 1 3 i 1 3Created by T. Madas)
Created by T. MadasQuestion 17The complex number z lies in the region R of an Argand diagram, defined by theinequalitiesπ3 arg ( z 4 ) πand0 arg ( z 12 ) 5π.6a) Sketch the region R , indicating clearly all the relevant details.The complex number w lies in R , so that w is minimum.b) Find w , further giving w in the form u iv , where u and v are real numbers.w 3 , w 3 3 3iCreated by T. Madas
Created by T. MadasQuestion 18The point P represents the number z x iy in an Argand diagram and further satisfiesthe equation 1 iz πarg , z i . 1 z 4Use an algebraic method to find an equation of the locus of P and sketch this locusaccurately in an Argand diagram.x 2 y 2 1, such that y x 1Created by T. Madas
Created by T. MadasQuestion 19The complex number x iy in the z plane of an Argand diagram satisfies the inequalityx2 y2 x 0 .a) Sketch the region represented by this inequality.A locus in the z plane of an Argand diagram is given by the equation z 1 πarg . z 4b) Sketch the locus represented by this equation.sketchCreated by T. Madas
Created by T. MadasQuestion 20The complex number z satisfies the relationshiparg ( z 2 ) arg ( z 2 ) π4.Show that the locus of z is a circular arc, stating the coordinates of its endpoints. the coordinates of its centre. the length of its radius.( 2, 0 ) , ( 2,0 ) , ( 0, 2 ) ,Created by T. Madasr 2 2
Created by T. MadasCOMPLEX FUNCTIONSCreated by T. Madas
Created by T. MadasQuestion 1A transformation from the z plane to the w plane is defined by the complex functionw 3 z, z 1 .z 1The locus of the points represented by the complex number z x iy is transformed tothe circle with equation w 1 in the w plane.Find, in Cartesian form, an equation of the locus of the points represented by thecomplex number z .x 1Question 2Find an equation of the locus of the points which lie on the half line with equationarg z π4, z 0after it has been transformed by the complex function1w .zarg w Created by T. Madasπ4
Created by T. MadasQuestion 3The complex functionw 1, z 1, z » , z 1z 1transforms the point represented by z x iy in the z plane into the point representedby w u iv in the w plane.Given that z satisfies the equation z 1 , find a Cartesian locus for w .u 12Created by T. Madas
Created by T. MadasQuestion 4The complex function w f ( z ) is given byw 3 zwhere z », z 1 .z 1A point P in the z plane gets mapped onto a point Q in the w plane.The point Q traces the circle with equation w 3 .Show that the locus of P in the z plane is also a circle, stating its centre and its radius.()centre 3 ,0 , radius 322Created by T. Madas
Created by T. MadasQuestion 5The general point P ( x, y ) which is represent by the complex number z x iy in thez plane, lies on the locus ofz 1.A transformation from the z plane to the w plane is defined byw z 3, z 1 ,z 1and maps the point P ( x, y ) onto the point Q ( u , v ) .Find, in Cartesian form, the equation of the locus of the point Q in the w plane.u 2Created by T. Madas
Created by T. MadasQuestion 6The point P represented by z x iy in the z plane is transformed into the point Qrepresented by w u iv in the w plane, by the complex transformationw 2z, z 1.z 1The point P traces a circle of radius 2 , centred at the origin O .Find a Cartesian equation of the locus of the point Q .(Created by T. Madasu 83)2 v 2 169
Created by T. MadasQuestion 7The complex numbers z x iy and w u iv are represented by the points P and Q ,respectively, in separate Argand diagrams.The two numbers are related by the equationw 1, z 1 .z 1If P is moving along the circle with equation( x 1)2 y 2 4 ,find in Cartesian form an equation of the locus of the point Q .u 2 v2 14Created by T. Madas
Created by T. MadasQuestion 8A transformation from the z plane to the w plane is defined by the equationw z 2i, z 2.z 2Find in the w plane, in Cartesian form, the equation of the image of the circle withequation z 1 , z » .2(u 13 ) ( v 43 )2 89Question 9A transformation from the z plane to the w plane is given by the equationw 1 2z, z 3.3 zShow that the in the w plane, the image of the circle with equation z 1 , z » , isanother circle, stating its centre and its radius .(u 85 )2( ) v 2 49 , centre 5 ,0 , r 76488Created by T. Madas
Created by T. MadasQuestion 10The complex numbers z x iy and w u iv are represented by the points P and Q ,respectively, in separate Argand diagrams.The two numbers are related by the equation1w , z 0.zIf P is moving along the circle with equationx2 y2 2 ,find in Cartesian form an equation for the locus of the point Q .u 2 v2 12Created by T. Madas
Created by T. MadasQuestion 11The complex numbers z x iy and w u iv are represented by the points P and Qon separate Argand diagrams.In the z plane, the point P is tracing the line with equation y x .The complex numbers z and w are related byw z z2 .a) Find, in Cartesian form, the equation of the locus of Q in the w plane.b) Sketch the locus traced by Q .v u 2u 2 or y x 2 x 2Created by T. Madas
Created by T. MadasQuestion 12The complex numbers z x iy and w u iv are represented by the points P and Qon separate Argand diagrams.In the z plane, the point P is tracing the line with equation y 2 x .Given that he complex numbers z and w are related byw z2 1find, in Cartesian form, the locus of Q in the w plane.4u 3v 4 or 4 x 3 y 4Created by T. Madas
Created by T. MadasQuestion 13A transformation of the z plane to the w plane is given byw 1 3z, z » , z 1,1 zwhere z x iy and w u iv .The set of points that lie on the y axis of the z plane, are mapped in the w plane onto acurve C .Show that a Cartesian equation of C is( u 1)2 v 2 4 .proofCreated by T. Madas
Created by T. MadasQuestion 14The complex function w f ( z ) is given by1w , z » , z 0 .zThis function maps a general point P ( x, y ) in the z plane onto the point Q ( u, v ) in thew plane.Given that P lies on the line with Cartesian equation y 1 , show that the locus of Q isgiven by11w i .22proofCreated by T. Madas
Created by T. MadasQuestion 15A transformation of the z plane onto the w plane is given byw az b, z » , z cz cwhere a , b and c are real constants.Under this transformation the point represented by the number 1 2i gets mapped to itscomplex conjugate and the origin remains invariant.a) Find the value of a , the value of b and the value of c .b) Find the number, other than the number represented by the origin, which remainsinvariant under this transformation.a 52 , b 0 , c 52 , z 5Created by T. Madas
Created by T. MadasQuestion 16A transformation of the z plane to the w plane is given byw 1, z » , z 2z 2where z x iy and w u iv .The line with equation2x y 3is mapped in the w plane onto a curve C .a) Show that C represents a circle and determine the coordinates of its centre andthe size of its radius.The points of a region R in the z plane are mapped onto the points which lie inside Cin the w plane.b) Sketch and shade R in a suitable labelled Argand diagram, fully justifying thechoice of region.centre at 1, 1 , radius 522(Created by T. Madas)
Created by T. MadasQuestion 17A transformation of the z plane to the w plane is given byw z2 , z » ,where z x iy and w u iv .The line with equation y 1 is mapped in the w plane onto a curve C .Sketch the graph of C , marking clearly the coordinates of all points where the graph ofC meets the coordinate axes.sketchCreated by T. Madas
Created by T. MadasQuestion 18A transformation of points from the z plane onto points in the w plane is given by thecomplex relationshipw z2 , z » ,where z x iy and w u iv .Show that if the point P in the z plane lies on the line with equationy x 1,the locus of this point in the w plane satisfies the equationv 1 2u 1 .2()proofCreated by T. Madas
Created by T. MadasQuestion 19A complex transformation from the z plane to the w plane is defined byw z i, z » , z 3i .3 izThe point P ( x, y ) is mapped by this transformation into the point Q ( u, v ) .It is further given that Q lies on the real axis for all the possible positions of P .Show that the P traces the curve with equationz i 2.proofCreated by T. Madas
Created by T. MadasQuestion 20A transformation of the z plane to the w plane is given byw 2z 1, z » , z 0zwhere z x iy and w u iv .The circle C1 with centre at 1, 1 and radius 5 in the z plane is mapped in the w22plane onto another curve C2 .()a) Show that C2 is also a circle and determine the coordinates of its centre and thesize of its radius.The points inside C1 in the z plane are mapped onto points of a region R in the wplane.b) Sketch and shade R in a suitably labelled Argand diagram, fully justifying thechoice of the region.( )centre at 3 , 0 , radius 122Created by T. Madas
Created by T. MadasQuestion 21A transformation of the z plane to the w plane is given by1w z , z » , z 0 ,zwhere z x iy and w u iv .The locus of the points in the z plane that satisfy the equation z 2 are mapped in thew plane onto a curve C .By considering the equation of the locus z 2 in exponential form, or otherwise, showthat a Cartesian equation of C is36u 2 100v 2 225 .proofCreated by T. Madas
Created by T. MadasQuestion 22A transformation from the z plane to the w plane is defined by the equationw i z 1, z » .Sketch in the w plane, in Cartesian form, the equation of the image of the half line withequationarg ( z 2 ) π4, z » .graphCreated by T. Madas
Created by T. MadasQuestion 23A transformation from the z plane to the w plane is defined by the equationf (z) iz, z » .z iFind, in Cartesian form, the equation of the image of straight line with equationz i z 2 , z » .2(u 25 ) ( v 45 )Created by T. Madas2 15
Created by T. MadasQuestion 24The complex function w f ( z ) is given byw 1, z 1.1 zThe point P ( x, y ) in the z plane traces the line with Cartesian equationy x 1.Show that the locus of the image of P in the w plane traces the line with equationv u.proofCreated by T. Madas
Created by T. MadasQuestion 25The complex function w f ( z ) satisfies1w , z » , z 0 .zThis function maps the point P ( x, y ) in the z plane onto the point Q ( u, v ) in the wplane.It is further given that P traces the curve with equation11z i .22Find, in Cartesian form, the equation of the locus of Q .v 1Created by T. Madas
Created by T. MadasQuestion 26z cosθ isin θ , π θ π .a) Show clearly that2θ 1 i tan .1 z2The complex function w f ( z ) is defined byw 2, z » , z 1 .1 zThe circular arc z 1 , for which 0 arg z π2, is transformed by this function.b) Sketch the image of this circular arc in a suitably labelled Argand diagram.proof/sketchCreated by T. Madas
Created by T. MadasQuestion 27The complex function with equationf (z) 1z2, z » , z 0maps the complex number x iy from the z plane onto the complex number u iv inthe w plane.The line with equationy mx , x 0 ,is mapped onto the line with equationv Mu ,where m and M are the respective gradients of the two lines.Given that m M , determine the three possible values of m .m 0, 3Created by T. Madas
Created by T. MadasQuestion 28A complex transformation of points from the z plane onto points in the w plane isdefined by the equationw z2 , z » .The point represented by z x iy is mapped onto the point represented by w u iv .Show that if z traces the curve with Cartesian equationy 2 2 x2 1 ,the locus of w satisfies the equationv 2 4 ( u 1)( 2u 1) .proofCreated by T. Madas
Created by T. MadasQuestion 29The complex function w f ( z ) is defined byw The half line with equation arg z π41, z » , z 1.z 1is transformed by this function.a) Find a Cartesian equation of the locus of the image of the half line.b) Sketch the image of the locus in an Argand diagram.2(u 12 ) ( v 12 )Created by T. Madas2 1 , v 0, u 2 v 2 u 02
Created by T. MadasQuestion 30The complex function w f ( z ) is defined byw The half line with equation arg z 3z i, z » , z 1.1 z3πis transformed by this function.4a) Find a Cartesian equation of the locus of the image of the half line.b) Sketch the image of the locus in an Argand diagram.( u 1)2 ( v 1)2 5,Created by T. Madasv 1 u 13
Created by T. MadasCOMPLEX SERIESCreated by T. Madas
Created by T. MadasQuestion 1The following convergent series C and S are given by111C 1 cos θ cos 2θ cos3θ .248111S sin θ sin 2θ sin 3θ .248a) Show clearly thatC iS 2.2 eiθb) Hence show further thatC 4 2cos θ,5 4cos θand find a similar expression for S .S Created by T. Madas2sin θ5 4cos θ
Created by T. MadasQuestion 2The following finite sums, C and S , are given byC 1 5cos 2θ 10cos 4θ 10cos 6θ 5cos8θ cos10θS 5sin 2θ 10sin 4θ 10sin 6θ 5sin 8θ sin10θ5By considering the binomial expansion of (1 A ) , show clearly thatC 32 cos5 θ cos 5θ ,and find a similar expression for SS 32cos5 θ sin 5θCreated by T. Madas
Created by T. MadasQuestion 3The following convergent series S is given belowS sin θ 1 sin 2θ 1 sin 3θ 1 sin 4θ .3927By considering the sum to infinity of a suitable geometric series involving the complexexponential function, show thatS 9sin θ.10 6cos θproofCreated by T. Madas
Created by T. MadasQuestion 4The sum C is given below n n n nC 1 cos θ cos θ cos 2 θ cos 2θ cos3 θ cos3θ . ( 1) cos n θ cos nθ 1 2 1 Given that n » determine the 4 possible expressions for C .Give the answers in exact simplified form.n 4k , k » : C cos nθ sin n θ , n 4k 1, k » : C sin nθ sin n θ ,n 4k 2, k » : C cos nθ sin n θ , n 4k 3, k » : C sin nθ sin n θCreated by T. Madas
Created by T. MadasQuestion 5The following convergent series S is given belowS sin θ sin 2θ sin 3θ sin 4θ .1!2!3!4!By considering the sum to infinity of a suitable series involving the complex exponentialfunction, show thatS e cosθ sin ( sin θ ) .proofCreated by T. Madas
sin5 sin 16cos 12cos 1θ θ θ θ (4 2). a) Use de Moivre's theorem to prove the validity of the above trigonometric identity. Consider the general solution of the trigonometric equation sin5 0θ . b) Find exact simplified expressions for cos 2 5 π and cos 2 2 5 π , fully justifying each step in the workings. cos 2 3 5 5 8 π
