Question

and z2 = 1 1 + 3i 3-i a) Given that zı = find z such that z = 2 + i 4- ¿ 22 Give your answer in the form of a + bi. Hence, find the modulus and argument of z, such that -- < arg(2) < 7. (6 marks) b) Given w = = -32, i. express w in polar form. (1 marks) ii. find all the roots of 2b = -32 in the form of a +ib. Show all he roots on an Argand diagram. (6 marks) c) Use de Moivre's theorem to show that cos 40 = 8 cos 0 - 8 cos0 +1. Hence solve 163-16x2 + 3 = 0. (7 marks)

Answer 1

W) 2,= 173 Z2 = 3-1 2+i (143 y- 3-i . (1-9461)(4-1) 6-2i+zit Zz 27 gi36-32 +81+2uit6 6+ 7+ -26+32i (-26+32; 32i) (7-3) 7+ (7+) (7-8) -182 +261 +22418+ 32 - 150 +250i 50 50 E-3+5i Iz1 = f(23) +(02) = 34 ange = tant (1) = 1.0303 6) W: - 32 3 w52 (4480+isin of wz 32 (ca) + isin) 32,0-5,9=5 Z. 32 +zi 8 - 5+1 +2115+ 5132 [cos (T+27:9 tisin T+ 25:0) sin T+ 27.0)] = + 1 *+1) 81882 (cofit 1,217) +isin Ty29)= -15 K-2,40 ) simptor)] = -2 * = 3, 852 [cmmt br) + i sid 467)) = -57 -21 4545 Kih, 5/32 risin T t 2")] - 1418-2185 CASTS + 80 are all room of 2 -32 of

N А. о 2 Во

C) (cesetisino)' = cos 40 + i sin 40 -) cosunget is in yox= cos a+ucos? (isinx) + 6eos (ision)? + y cosx (isinaß tri sina ja costantuicos sinnt si costusnaut his corn sinn tin sinan = (cos" n-6 cos?nsin ²x+ sinx) +i (4 cosBusion using a Equating the real part we get casso cosuna costa-6consin Be+sinin = con csinan ( co Ye - Ecosu (1-сog?n) +(1-CO2x) Gos-bosbe to Gost 1 year) + 6- co2x² + 4 least (as)" = 7toxt-tocona + bca na 400² = cos n-600g2n + Google + It cosu-2 con cocoshu= 8 costu - 8 cos2x+1 2 16x 162 ²13-218048 x ² + 1) +10 - 2 Count 1 = 0 costantelor 글 16x9_164² +3=0 = 1684-12x² 4 2 ² +3=0 >) 4² (4x²-3) -1 (4n²-3) = 0 =) (un ²-1) (4x2-3) = 0 3) 2=+3,+ 2