cos The points in an Argand diagram which represent the numbers (iii) Write down a polynomial equation of degree 5 which is satisfied by w.

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1 FP3 Complx Numbrs. Jun qu.3 In this qustion, w dnots th complx numbr cos + i sin. 5 5 Exprss w, w 3 and w* in polar form, with argumnts in th intrval θ <. [] Th points in an Argand diagram which rprsnt th numbrs, + w, + w + w, + w + w + w 3, + w + w + w 3 + w ar dnotd by A, B, C, D, E rspctivly. Sktch th Argand diagram to show ths points and join thm in th ordr statd. (Your diagram nd not b xactly to scal, but it should show th important faturs.) [] Writ down a polynomial quation of dgr 5 which is satisfid by w. []. Jun qu.5 Convrgnt infinit sris C and S ar dfind by C = + cos θ + cos θ + 8 cos 3θ +..., S = Show that C + is = sin θ + sin θ + sin 3θ [] Hnc show that C = cosθ, and find a similar xprssion for S. [] 5 cosθ 3. Jan qu. Writ down, in cartsian form, th roots of th quation z =. [] Hnc solv th quation w = ( w), giving your answrs in cartsian form. [5]. Jan qu. 7 Solv th quation cos θ =, for < θ <. [3] By using d Moivr s thorm, show that cos θ ( cos θ )( cos θ cos θ + ). [5] 5 7 cos cos cos cos Hnc find th xact valu of, justifying your answr. [5]

2 5. Jun 9 qu. Find th cub roots of 3 + i, giving your answrs in th form cos θ + i sin θ, whr θ <. []. Jun 9 qu. It is givn that th st of complx numbrs of th form r for < θ and r >, undr multiplication, forms a group. Writ down th invrs of 3 i 5. [] Prov th closur proprty for th group. [] Z dnots th lmnt iγ, whr < γ <. Exprss Z in th form, whr < θ <. [] 7. Jun 9 qu.7 Us d Moivr s thorm to prov that tan 3θ tanθ (3 tan θ ). [] 3 tan θ (a) By putting θ = in th idntity in part, show that tan is a solution of th quation t 3 3t 3t + =. [] (b) Hnc show that tan = 3. [] Us th substitution t = tan θ to show that 3 t(3 t ) ( 3t )( + t dt = a ln b, ) whr a and b ar positiv constants to b dtrmind. [5] 8. Jan 9 qu. 3 + i Exprss in th form r, whr r > and θ <. [3] 3 i Hnc find th smallst positiv valu of n for which 3 + i 3 i n is ral and positiv. [] 9. Jan 9 qu. 8 By xprssing sin θ in trms of and, show that sin θ 3 (cos θ cos θ + 5 cos θ ). [5] Rplac θ by ( θ) in th idntity in part to obtain a similar idntity for cos θ. [3] Hnc find th xact valu of (sin θ cos θ ) dθ. []

3 . Jun 8 qu. By xprssing cosθ in trms of and, show that cos 5 θ (cos 5θ + 5cos3θ + cosθ). [5] Hnc solv th quation cos 5θ + 5cos3θ + 9cosθ = for θ. []. Jun 8 qu.7 Th roots of th quation z 3 = ar dnotd by, ω and ω. Sktch an Argand diagram to show ths roots. [] Show that + ω + ω =. [] Hnc valuat (a) ( + ω)( + ω ), [] (b) +. + ω + ω [] (iv) Hnc find a cubic quation, with intgr cofficints, which has roots, + ω and. + ω []. Jan 8 qu. Th intgrals C and S ar dfind by C = x cos3x dx and S = By considring C + is as a singl intgral, show that x sin 3x dx. C = ( + 3 ), and obtain a similar xprssion for S. 3 (You may assum that th standard rsult for kx ( a + ib) x ( a + ib) x constant, so that dx = a + ib 3. Jan 8 qu. 7 dx rmains tru whn k is a complx [8] (a) Vrify, without using a calculator, that θ = is a solution of th quation 8 sin θ = sin θ. [] (b) By sktching th graphs of y = sin θ and y = sin θ for θ or othrwis, find th othr solution of th quation sin θ = sin θ in th intrval < θ <. []

4 Us d Moivr s thorm to prov that sin θ sin θ( cos θ cos θ + 3). [5] Hnc show that on of th solutions obtaind in part satisfis cos ( θ = ), and justify which solution it is. [3]. Jun 7 qu. By writing z in th form r, show that zz* = z. [] Givn that zz* = 9, dscrib th locus of z [] 5. Jun 7 qu.5 Us d Moivr s thorm to prov that cos θ = 3 cos θ 8 cos θ + 8 cos θ. [] Hnc find th largst positiv root of th quation x 9x + 3x 3 =, giving your answr in trigonomtrical form. []. Jun 7 qu.7 Show that (z iφ )(z iφ ) z (cosφ) z +. [] Writ down th svn roots of th quation z 7 = in th form and show thir positions in an Argand diagram. [] Hnc xprss z 7 as th product of on ral linar factor and thr ral quadratic factors. [5] 7. Jan 7 qu. 3 Solv th quation z z + 3 =, and giv your answrs in th form r(cosθ ± isinθ), whr r > and θ. [] Givn that Z is ithr of th roots found in part, dduc th xact valu of Z 3. [3] 8. Jan 7 qu. 8 Us d Moivr s thorm to find an xprssion for tan θ in trms of tan θ. [] Dduc that cot θ = cot θ cot θ + 3. [] cot θ cotθ Hnc show that on of th roots of th quation x x + = is cot. [3] (iv) Hnc find th valu of cosc + cosc 3, justifying your answr. [5]

5 9. Jun qu. (a) Givn that z = i and z = i 3, xprss z z and z z in th form r, whr r > and θ <. [] (b) Givn that w = (cos 8 + i sin 8 ), xprss w 5 in th form r(cos θ + i sin θ), whr. Jun qu.7 r > and θ <. [3] Th sris C and S ar dfind for < θ < by C = + cos θ + cos θ + cos 3θ + cos θ + cos 5θ, S = sin θ + sin θ +sin 3θ + sin θ + sin 5θ. Show that C + is = [] Dduc that C = sin 3θ cos 5 θ cosc θ and writ down th corrsponding xprssion for S. [] Hnc find th valus of θ, in th rang < θ <, for which C = S. []. Jan qu. By xprssing cos θ and sin θ in trms of and, or othrwis, show that cos θ sin θ = (cos θ cos θ cos θ + ) [] 3 Hnc find th xact valu of 3 cos θ sin θ dθ. [3]. Jan qu. 5 Solv th quation z = (cos + i sin), giving your answr in polar form. [] By writing your answr to part in th form x + iy, find th four linar factors of z +. [] Hnc, or othrwis, xprss z + as th product of two ral quadratic factors. [3]