COMPLEX NUMBER
FE /Maths-I/ GQ/Complex number COMPLEX NUMBER 1. The number of the form x + i y where x and y are real numbers and i = -1 i. e.(i 2 1) is called a complex number and it is denoted by z i.e . z = x + i y.The complex number z is represented by a point P(x, y) on XY - plane 2. x is called real part and is denoted Re z = x & y is called imaginary part and is denoted by Im z y. 3. If z = x + i y be any complex number then its conjugate is denoted by z and z x iy 5. Euler' s Formulae i) ei cos i sin 6. bg bg Polar form b ii) e- i cos i sin g z = r cos + i sin r ei 7. r = x 2 y 2 is said to be modulus of the complex number z and is denoted by z x iy 8. b g x 2 y 2 , tan 1 y / x Where r = x2 y2 b g = tan -1 y / x is said to be amplitude or argument of the complex number b g z = x + iy and is denoted by arg z = amp z tan 1 y / x The value such that - is said to be principle value of the argument. 9. Properties of modulus and argument 1. z1 z2 z1 z2 2. z1 z2 3. z1z2 z1 z2 z1 - z 2 z z1 1 z2 z2 z 6. arg 1 arg z1 arg z2 z2 4. b g 5. arg z1z2 arg z1 arg z 2 Exponential form of circular function: e i e i e i e i i sin = ii cos = 2i 2 Hyperbolic Functions ex e x 1. Let x be real or complex then is called hyperbolic sine of x and is denoted by sinh x 2 ex e x and is called hyperbolic cosine of x and is denoted cosh x 2 ex e x ex e x sinh x = ; coshx = 2 2 x x sinh x e e 1 2 tanh x = x sech x = x x cosh x e e cosh x e e x x x cosh x e e 1 2 coth x = x cosech x = = x x x sinh x e e sinh x e e bg bg Relations between circular function and hyperbolic functions b g b g b g (i) Sin i x = i sinh x (iii) tan i x = i tanh x (v) Cosh i x = cos x b g b g b g (ii) cos i x = cosh x (iv) Sinh i x = i sin x (vi) tanh i x = i tan x 1 FE /Maths-I/ GQ/Complex number 3. Hyperbolic Identities. (i) sinh (-x) = - sinh x (iii) tanh -x = - tanh x ( ii) cosh (-x) = cosh x (iv) e x cosh x sinh x b g 4. 5. Square Relations (i) Cosh2 - sin h2 x = 1 (ii) sech2 x + tanh2 x = 1 g Product Formulae (i) sinh ( A + B) + sinh (A - B) = 2 sinh A cosh B. (ii) sinh (A + B) - sinh (A - B) = 2 cosh A sinh B (iii) cosh (A + B) + cosh (A - B) = 2 cosh A cosh B (iv) cosh (A + B) - cosh (A - B) = 2 sinh A sinh B C+D C-D (v) sinh C + sinh D = 2 sinh cosh 2 2 C+D C-D (vi) sinh C - sinh D = 2 cosh sinh 2 2 C+D C-D (vii) cosh C + cosh D = 2 cosh cosh 2 2 C+D C-D (viii) cosh C - cosh D = 2 sinh sinh 2 2 FG H FG H 7. FG H FG H IJ K IJ K IJ K IJ K FG H FG H IJ K IJ K FG IJ H K FG IJ H K Formulae for 2x and 3x ii) cosh 2 x = cosh 2 x sinh 2 x 1 iv) sinh 2 x cosh 2 x 1 2 i) sinh 2 x = 2 sinh x cosh x 1 iii) cosh 2 x cosh 2 x 1 2 2 tanh x v) tanh 2 x = 1 + tanh 2 2 x b b g ix) sinh 2 x = g vi) sinh 3x = 3 sinh x + 4 sinh 3 x vii) cosh 3x = 4 cosh 3 x 3 cosh x 8. (iii) coth2 x - cosech2 x = 1 Addition formulae Sinh (A B) = sinh A cosh B cosh A sinh B Cosh (A B) = coshA coshB sinh A sinh B tanh A tanh B tanh A B 1 tanh A tanh B b 6. (v) e -x cosh x sin h x viii) tanh 3x = 2 tanh x x) cosh 2 x = 1 - tanh 2 x 3 tanh x + tanh 3 x 1 3 tanh 2 x 1 + tanh 2 x 1 tanh 2 x Values of Hyperbolic functions x sinh x coshx tanhx - - -1 0 0 1 0 1 EXAMPLES: 1. If x iy 2. If i 2 a 2 b2 then prove that x 2 y 2 2 c d2 c id a ib 1 a ib then prove that 2 2 a 2 b2 1 2 FE /Maths-I/ GQ/Complex number 1 x iy 3. If x 2 y 2 1 Then show that x iy 1 x iy 4. Express the complex number x 1 iy x 1 iy 2 2 number then show that x y 1 in the form a ib.If this is a purely imaginary 5. Find the complex numbers whose sum is 4 and whose product is 8. 6. Find the complex number z such that 2 i) arg z +1 and arg z -1 6 3 b g b g FG z + iIJ / 4 HzK iii) Find z if arg b z + 2ig = / 4 and arg b z - 2ig = ii) Find z if z + i z and arg 3 4 7. Find the loci represented by ii) z + 2i z - 2i = 6 ii) z - 3 z + 3 = 4 iii) z - 2 + i = 3 iv) z - 5- 6i = 4 DE MOIVRE’S THEROEM bCos 5 - i sin 5g bcos 7 + i sin 7g bCos 4 - i sin 4 g bcos + i sin g 3 4 1. Simplify 9 2. Express in the form a + ib 5 bCos 3 + i sin 3g FG Cos 3 + i sin 3 IJ H 2 2K 4 bcos 5 - i sin 5 g FG cos 4 isin 4 IJ H 5 5K 4 /5 2/9 10 b1 ig d1- i 3i 3. Express in the form a + ib b1 ig d1 i 3i b 1 ig d 3 i i 4. Express in the form a + ib b 1 i g d 3 i i 1 i 5. It z = Then simplify b z g d z i 2 2 d1 + i 3 i 6. Find the modulus and argument of d 3 i i 6 4 8 5 8 4 8 4 10 10 13 11 C.N. 1 1. Prove that LM1 + sin + i cos OP N 1 + sin - i cos Q Evaluate LM1 + sin + i cos OP N 1 + sin - i cos Q 3. Show that LM1 + cos + i sin OP N 1 + cos - i sin Q 2. n cos FG n nIJ H2 K i sin FG n nIJ H2 K n n cosn + i sinn 3 FE /Maths-I/ GQ/Complex number b1+ cos + i sin g b1 + cos n 4. Show that b 5. Show that 1 + sin + i cos 6. Show that bcos g + b1 n b g 7. Show that g n = 2 n+1 cosn + sin - i cos g n 2 cos n 2 FG H4 - = 2 n+1 cosn bcos - cos g - i bsin FG - IJ cos n FG + + IJ H2K H 2 K - cos + i sin - sin = 2 n+1 sin n - i sin g n + IJ 2K - sin cos g FG n H4 - n 2 IJ K n F I i G - J 1 + cos + i sin cot / 2 e H 2 K 1 - cos + i sin b 8. If 1+ cos + i sin i) v = u tan 3 2 g b1 + cos2 g + i sin2 u + iv ii) u 2 v 2 16cos2 cos2 then prove that 2 9. Prove that a = cos + i sin , b = cos + i sin then show that (a + b) (ab -1) sin sin (a - b) (ab +1) sin sin 10. If a = cos + i sin , b = cos + i sin , c = cos + i sin then prove that ba + b g b b + cg bc + a g abc = 8 cos FG - IJ cos FG - IJ cos FG - IJ H 2K H2K H 2K 11. If a = cos 2 + i sin 2 , b = cos 2 i sin 2 and c = cos 2 , + i sin 2 then prove that b ab c c = 2 cos + - ab g 12. If a = cos 3 + i sin 3 , b = cos 3 i sin 3 and c = cos 3 , + i sin 3 then prove that 3 ab c 3 b c = 2 cos + - ab g C.N.2 1. Prove that (4n) th power of 1+ 7i is equal to (-4) n where n is positive integer (2 - i) 2 2. If z = cos + i sin then prove that i) 2 1 i tan 1+ z 2 3. If z1 e i and z 2 e i then prove that b g sin - 4. If a 2 + b 2 c 2 1 & b ic (1 a)z then prove that 1 2i ii) FG z Hz 1 1+ z i cot 1- z 2 2 z2 z1 IJ K a + ib 1 iz 1+ c 1 iz 2 5. If sin itan then prove that cos isin 1 tan 2 6. If sin itan then prove that cos isin tan 4 2 1 tan FG H IJ K 4 FE /Maths-I/ GQ/Complex number C.N .3 1. If z1 and z 2 are any two complex numbers such that z z z1 z 2 and z1 z 2 then show that 1 2 z1 z 2 is purely imaginary 2. If z1 and z 2 are any two complex numbers then prove that z1 z 2 2 z1 z 2 2 2 2 z1 2 z2 3. Prove that the statement Rez 0 & z 1 z 1 are equivalent where z x iy 4. If z1 and z 2 are any two complex numbers such that z1 + z 2 then show that argz1 argz 2 5. Prove that z1 z 2 and 2 z 1 arg z z C.N. 4 1. If z1 cos + i sin z 2 cos + i sin where 0 < < / 2 0< /2 1+ z12 1 i z1z 2 Find the modulus and argument of 2. Prove that e 2 a i cot -1b LM bi -1 OP N bi +1 Q a 1 b g b g b g.sinn.cosec h c1 e h 3. If 1 i, 1 i & cot x 1 prove that x x 4. Prove that b g c 1+ cosec / 2 1 e i 1 2 n n n i 1 2 C.N. 5 1. If and are the roots of the equation x 2 2 3 x + 4 = 0 Then prove that 3 3 0 FG n IJ H4K 2. If and are the roots of the equation x 2 - 2x + 2 = 0 Then prove that n n 2.2 n/2 cos hence deduce that 8 8 32 3. If and are the roots of the equation n n 2 n 1 cos FG n IJ H 3K x 2 2 x 4 0 Then prove that Hence find the value of 6 + 6 4. If and are the roots of the equation x 2 3x 1 0 then prove that n n 2 cos FG n IJ H 6K Hence prove that 12 12 2 5. If and are the roots of the equation z 2 sin 2 - z sin 2 + 1 = 0 then prove that n n 2 cos n cosec n ) 6. If z = -1+ i 3 & n is integer, then prove that z 2n 2 n z n 2 2n 0 if n is not a mutiple of 3. 7. If x + 1 1 = 2 cos , prove that x r r 2 cos r x x 5 FE /Maths-I/ GQ/Complex number 1 1 1 = 2 cos and y + 2 cos then prove that one of the values of x m y n m n x y x y 8. If x + b b g xm yn is 2cos m + n and that of n + m is 2 cos m - n y x 9. If x + g 1 1 1 = 2cos, y 2 cos and z + 2 cos x y Z b b g 1 xm yn then prove that i) xyz + = 2 cos + + ii) n + m = 2 cos m - n xyz y x 1 1 1 10. If x = 2 sin , y 2 sin & z - 2 sin then prove that x y Z b 1 = 2 cos + + xyz i) xyz + g m ii) the one of n FG H x ny + m is 2 cos m n y x g IJ K C.N.6 1. If n is a positive integer then show that ba - i b g = 2 ea + b j cos LMN n tan FGH ba IJK OPQ Hence find the value of b 1 + i g + b 1 - i g Lm O 2. Prove that ba + ibg ba ibg 2 ea b j cos M tan b b / a gP Nn Q Hence find the value of b3 + 4ig b3- 4ig L n O 3. Prove that d1+ i 3 i d1 i 3 i 2 cos M P N3Q F -1+ i 3 I F 1 i 3 I RS1 if n = 3k ! Where k is a positive integer 4. Prove that G H 2 JK GH 2 JK T 2 if n = 3k ba + ibg n n + 2 2 n2 n m n n m n m 2 2n 2 2/3 n n -1 -1 2/3 n 1 n n C.N. 7 FG IJ H2 K 1. If x r cos r + i sin FG IJ H2 K then prove that r i) x1 x 2 x 3 ............... ad = -1 ii) x 0 x1 x 2 x 3 ........... ad 1 2. If x r cos FG IJ i sin FG IJ H3 K H3 K r the prove that r i) x1 x 2 x 3 .......... ad = i b i) ca ii) x 0 x1 x 2 x 3 ........... ad i g ba ib g ......ba ib g = A + iB then prove that b h ca b h ........ ca b h A B Fb I b b F B IJ ii) tan + tan .......+tan G J tan G HAK a a Ha K 3. If a 1 ib1 2 1 2 1 -1 2 2 2 1 1 b gb 2 n 2 2 -1 2 n 2 2 gb g n 2 n 2 -1 n 2 1 n 4. If 1+ ia 1+ ib 1+ ic p + iq then prove that p tan tan -1a tan 1 b tan 1 c q b g bcos 2 + i sin2g.............. bcos n + i sin ng 1 4k then prove that = where K is any integer. nb n +1g 5. If cos + isin 6 FE /Maths-I/ GQ/Complex number b1+ xg p + p x p x ...... p x then prove that F n I F n I cosG J 2. p p p ........... 2 sinG J H4K H4K F n I 2 cosG J H4K n 6. If n is a positive integer and 1. p 0 p 2 p 4 ........... 2 n 2 2 0 1 n 2 n n2 1 3 5 n 1 2 3. p 0 p 4 p8 ......... 2 n 2 C.N. 8 1. If x = e i , y = e i and z = e i and x + y + z = 0 then prove that 2. If sin + sin = 0 and cos + cos = 0 then prove that b i. cos2 + cos 2 = 2 cos + + g 1 1 1 + + = 0 x y z b ii. sin2 + sin2 = 2sin + + g 3. If cos + cos + cos = 0 and sin + sin + sin = 0 then prove that i. cos2 + cos2 + cos2 = 0 ii. sin2 + sin2 + sin2 = 0 4. If cos + cos + cos = sin + sin + sin = 0 i. cos cos cos 3 2 2 2 b g 2 b g b g iii. cos + cos cos 0 then prove that ii. sin 2 sin 2 sin 2 = 3 / 2 b b g g b g iv. sin + sin sin 0 5. If a cos + b cos + c cos = 0 and a sin + b sin + c sin = 0 then prove that b i. a 3 cos 3 + b 3 cos 3 + c 3 cos3 = 3abc cos + + b ii. a 3 sin 3 + b 3sin3 + c 3 sin 3 3abc sin + + g g 6. If cos + cos + cos = sin + sin + sin = 0 then prove that b i. cos3 + cos3 + cos3 = 3 cos + + b g g i. cos3 + 8 cos3 + 27 cos3 = 18 cos b + + g ii. sin 3 + 8 sin3 + 27 sin3 = 18 sin b + + g ii. sin3 + sin3 + sin3 = 3 sin + + 7. If cos + 2 cos + 3 cos = sin + 2 sin + 3 sin 0 then prove that C.N. 9 Expansion of cosn , sinn in powers of sin , cos 1. Using De - Moivre' s theorem show that sin 3 = 3cos2 sin sin 3 2. Use De - Moiver's theorem to prove the following result cos 4 = cos4 6 cos2 sin 2 sin 4 3. Obtain the expansions of cos 5 and sin 5 in terms of powers of cos and sin 4. Prove that sin5 = 5 sin - 20 sin 3 +16 sin5 5. Prove that cos5 = 5 cos - 20 cos3 +16 cos5 6. Obtain the expansion of cos 6 in terms of powers of sin and cos 7. Prove that cos 6 = 1-18 sin 2 48 sin 4 32 sin 6 8. If sin6 = A cos5 sin - B cos3 sin 3 + C cos sin5 Find the values of A, B and C 7 FE /Maths-I/ GQ/Complex number 9. Prove that sin7 = 7cos6 sin - 35cos4 sin 3 21cos2 sin5 sin 7 sin5 16 cos4 12 cos2 + 1 sin sin7 11. Prove that 7 - 56 sin 2 112 sin 4 65 sin 6 sin 10. Prove that 12. Prove that sin7 64 cos6 80 cos4 24 cos2 1 sin 13. Prove that tan 5 = 14. Prove that tan 7 = 5 tan -10 tan 3 tan5 1 10 tan 2 5 tan 4 Hence deduce that 5 tan 4 10 10 tan 2 10 1 0 7t - 35t 3 21 t 5 t 7 where t = tan 1 21t 2 35t 4 7 t 6 Hence deduce that 1- 21tan 2 35 tan 4 7 tan 6 0 14 14 14 C.N. 10 Expansion of cosn , sinn in terms of sines or cosines of multiples of 1 sin 5 5 sin 3 10 sin 16 Obtain the expansions of cos6 and sin 6 in terms of cosines of multiples of 1 Prove that cos6 sin 6 3 cos 4 5 8 1 Use De - Moivre's theorem to prove that cos6 sin 6 4 cos 6 15 cos 2 2 6 7 Prove that - 2 sin sin 7 7 sin 5 21 sin3 - 35sin 1. Prove that sin5 2. 3. 4. 5. 6. Express cos8 as a series in cosines of multiples of 7. Prove that cos8 sin8 1 cos 8 + 28 cos4 + 35 64 8. Express cos5 sin 3 as a series in sines of multiples of -1 9. Prove that cos5 sin 3 = 7 sin 8 + 2 sin6 - 2 sin 4 - 6 sin2 2 10. Prove that cos5 sin 7 = -1 sin 12 - 2 sin10 - 4 sin 8 + 10 sin 6 + 5sin4 - 20 sin2 211 11. Prove that - 212 cos6 x sin 7 x sin13x - sin11x - 6 sin9x + 6 sin7x + 15 sin5x - 15 sin3x - 20 sinx 12. Show that 25 sin 4 cos2 = cos6 - 2cos4 - cos2 + 2. 13. If sin 4 cos3 = A 1 cos A 3 cos 3 A 5 cos 5 A 7 cos 7 Prove that A 1 9A 3 25A 5 49 A 7 0 2 1+ cos7 x3 x2 2x 1 where x = 2 cos 1+ cos e 14. Prove that j b g c h 2 15. Using De - Moivre's theorem show that 2 1+ cos8 x 4 4 x 2 2 where x 2 cos 16. Use De - Moivre's theorem to show that b gc h 1+ cos 9 = 1+ cos 16cos4 8 cos3 12 cos2 4 cos 1 2 8 FE /Maths-I/ GQ/Complex number SUMATION OF SERIES sin sin 2 sin 3 2 sin ......... 2 3 2 2 2 5- 4 cos 1 1 1 9 - 3 cos x Prove that 1+ cosx + cos 2x + cos 3x + ............ = 3 9 27 10 - 6 cos x 2 x Find the sum of the series sin + x sin + Sin + 2 ........... 2! e 3 e5 -1 Prove that e cos cos 3 + cos 5 +......... = tan -1 cos cosech 3 5 2 1 2 1 3 x sin Prove that x sin x sin 2 + x sin 3........= tan -1 2 3 1+ x cos n(n 1) n(n 1)(n 2) Find the sum of the series n sin sin2 sin3 ...upto n terms 1.2 1.2.3 C.N. 11 Roots of Complex Number 1. Show that 2. 3. 4. 5. 6. b b g g b FG H g IJ K 1. Find all the values of i 2 3 and show that their continued product is -1 F 1+ 3 I and show that their continued product is 1. 2. Find all the values of G H 2 JK 3. Find all the values of b1+ ig and show that their continued product is 1+ i 4. Show that the continued product of all the values of b1+ ig is b1+ ig 5. Find the continued product of all the values of b1- ig 3/ 4 1/ 5 1/ 8 2/3 C.N. 12 Solve the following equation 1. x 6 i 0 2. x 3 i 0 3. x 7 x 4 x 3 1 0 4. x14 127 x 7 128 0 5. x 9 x5 x 4 1 0 6. x 9 x 6 x 3 1 0 7. x 9 8x 6 x 3 8 0 8. x 7 x 4 i x 3 1 0 b g 9. x 7 64 x 4 64 x 3 64 c 2 0 h 10. x10 11x5 10 0 11. x 4 x 2 1 x 3 x 12. x 4 x 3 x 2 x 1 0 13. x 4 x 2 1 0 C.N. 13 Roots of Complex Number 1. Find the cube roots of unity and show that they can be expressed as 1, , 2 bg 2. Prove that the n th roots of unity are in geometric progression & their sum is zero & product is -1 3. If is the complex cube root of unity then show that b1- g 6 n-1 27 4. Find the n th roots of -1 and show that they can be expressed as , 3 , 5 ....... 2 n1 Also find their continued product 5. Show that the roots of the equation x5 1 0 can be written as b gc Hence prove that 1- 1- 2 h c1- h c1- h 5 3 1, , 2 , 3 , 4 . 4 6 If , 2 , 3 ........ 6 are the roots of the equation x 7 1 0 then prove that b1- g c1- h c1- h c1- h c1- h c1- h 7 2 3 4 5 6 9 FE /Maths-I/ GQ/Complex number 7. Solve the equation x 1 0 and show that 5 cx 1h = bx -1g FGH x 5 2 IJ FG x KH + 1 5 2 x cos 2 2 x cos IJ K 3 1 5 C.N. 14 1. Prove that n a + bi + n a - bi has n real values and find those of b g b g 5 2. Show that the roots of the equation x -1 32 x 1 FG 2r IJ H5K F 2r IJ 5 4 cos G H5K 5 3 1+ i 3 + 3 1- i 3 are given by -3 + 4i sin x = where r = 0, 1, 2, 3, 4 3. Find the common roots of x 4 1 0 & x 6 i 0 4. Solve the equation x12 1 0 and find which of its roots satisfy the equation x 4 x 2 1 0 b g 1 for its real part. 2 6 5. Show that every root of 1+ x x 6 0 has b g b g 6 6 6. Show that the roots of x +1 x 1 0 are given by - i cot b g b g 7 LM b2p +1g OP, p = 0,1,2,3,4,5 N 12 Q 7 7. Show that the roots of the equation x +1 x 1 are given by icot FG k IJ , H7K k = 1,2,3 Why k 0? b g 5 8. Show that the roots of the equation 2z -1 z = FG IJ H K 1 i k cot , k = 1,2,3,4 4 4 5 32 z5 are given by k0 9. Find the three cube roots of 1- cos - i sin 10. Solve the equation z5 3 i C.N. 15 1. If 1+ 2i is a root of the equation x 4 3x 3 8 x 2 7 x 5 0 Find all other roots 2. If one of the roots of the equation x 4 6 x 3 15x 2 18 x 10 0 is 1+ i Find all other roots. C.N. 16 Hyperbolic Functions Seperate the real and imaginary parts of b i. sin x + iy g b ii. cos x + iy g b iii. sinh x + iy g b iv. cosh x + iy g b g b iv. tan x + iy v. tanh x + iy g C.N. 17 10 FE /Maths-I/ GQ/Complex number b g cosh nx sinh nx L1+ tanh x OP cosh 2nx sinh 2nx 2. Prove that M N 1- tanh x Q L1+ tanh x OP cosh 6x sinh 6x 3. Prove that M N 1- tanh x Q L cosh x + sinh x OP cosh 2nx sinh 2nx 4. Prove that M N cosh x - sinh x Q 1. Show that cosh x - sinh x n n 3 n 5. Prove that cosec hx + coth x = coth x / 2 6. If 5 sinh x cosh x 5 , find tanh x 7. Pr ove that 16 cosh5 x cosh 5x 5 cosh 3x 10 cosh x b g 8. If x = tanh -1 0.5 then prove that sinh 2x = 4 / 3 and cosh 2x = 5 / 3 9. If sin = tanhx Prove that tan = sinhx 10. Prove that 1 cosh 2 x (M'96 ) 1 11 11- cosh 2 x 1 sinh 2 x 1 11 11+ sinh 2 x 11. Prove that C.N. 18 b g 1. If cosh + i x i y then prove that 2 i. x y2 1 cosh 2 sinh 2 b g 2. If cos-1 x i y i ii. then prove that i. x sec y cos ec 1 2 2 2 b 2 g 3. If sin x + i y u i v 2 2 b ii. x 2 sec h 2 y 2 cos ech 2 1 then prove that i. u cos ec x v sec x 1 2 x2 y2 1 cos2 sin 2 2 ii. u 2 sec h 2 y + v 2 cos ech 2 y 1 g 4. If sinh a + i b x i y then prove that x y2 i. 1 sinh 2 a cosh 2 a y2 x2 ii. 1 sin 2 b cos2 b 2 C.N. 19 b 1. If tan + i g = x + i y then prove that i. x 2 + y 2 + 2x cot 2 = 1 b ii. x 2 + y 2 - 2 y coth 2 = -1 g 2. If tanh a + i b = x + i y then prove that 2 2 i. 1 + x + y = 2x coth 2a 3. If x + i y = tan 4. If tanh FG + H FG H6 i 6 IJ K + i IJ K ii. x 2 + y 2 + 2 y cot 2 b = 1 then prove that x 2 + y 2 + 2x 3 = x + i y then prove that x 2 + y 2 + = 1 2y = 1 3 11 FE /Maths-I/ GQ/Complex number C.N. 20 b g 1. If sin + i e i then prove that b g b g cos2 = sin 2. If sinh i e i then prove that sinh 4 cos2 cos4 3. If cosh + i cos i sin then prove that sin 2 sin 4 sinh 4 C.N. 21 1. If sin + i P cos + isin then prove that 1 i. P 2 cosh 2 - cos2 ii. tan = tanh cos 2 b b g b b g g g 2. If e z sin u + iv and z = x + iy then prove that 2e 2x = cosh 2v - cos2u b 3. If x + iy = cos + i g then express x and y in terms of and . Hence show that cos2 and c h cosh 2 are the roots of the equation. 2 x 2 y 2 1 + x 2 0 b g 4. If sin -1 i x iy then show that sin 2 x and cosh 2 y are the roots of the equation c h p 1 p + 2 0 2 2 2 b g 5. If cos + i cos i sin then prove that cos2 cosh 2 2 C.N. 22 1- 2 2 cos 2 x 1. If tan x + iy i then prove that 2 2 1 cosh 2 y x y c 2. If x + iy = c cot u + iv then prove that = sin 2u sinh 2v cosh 2v - cos2u sin 2x tan u 3. If tan x + iy = sin u + iv then prove that sinh 2y tanh v b g b b g b g C.N. 23 FG H 1. If + i = tanh x + 2. If cosec FG H4 IJ K + ix g i 4 IJ K then prove that 2 2 1 = u + iv then prove that FG H 3. If x + iy = 2 cosh + i 4 IJ K cu 2 v2 h 2 c = 2 u2 v2 h then prove that x 2 y 2 2 C.N. 24 b g 1. If fan + i tan i sec then prove that FG FG H H 2. If u = log tan 3. If tan + 4 2 IJ IJ KK i. 2 = n + 2 then prove that i. tanh u / 2 = tan ii. e 2 cot 2 ii. cosh u.cos 1 2 x u = tanh then prove that i. sinh u = tan x ii. coshu = secx 2 2 12 FE /Maths-I/ GQ/Complex number b g 4. If cos + i r ei then show that = LM b g OP N b gQ sin - 1 log 2 sin 5. If log tanx = y then prove that 1 tan n x cot n x 2 ii. cosh n +1 y + cosh n -1 y = 2 cosh ny.cosec 2x i. sinh ny = b g b g x = tan 2 2 iii. x = log tan + iv. 2 tan 1 (e x ) 4 2 2 6. If cosh x = sec then prove that i. tanh LM FG N H ii. sinhx = tan IJ OP KQ C.N. 25 x + iy - c 1. If = e u+iV where x, y, u and v are real then prove that x + iy + c -c sinh u c sin v i. x = ii. y = cosh u - cos v cosh u - cos v b g 2. If sin + i tan i sec then prove that cos2 cosh 2 = 3 b g 3. If tan / 4 + iv r e i then show that r = 1, tan = sinh 2v and tanh v = tan 2 4. If x = 2 sin cosh and y = 2 cos sinh then prove that 4x 4iy i. cosec - i + cosec + i = 2 ii. cosec - i - cosec + i = 2 2 x y x y2 b b g b g b g g 5. If x = 2cos cosh and y = 2 sin sinh then prove that b b g g 4x x y2 i. sec i + sec - i = b g bcosh v cos ug 2 b b g g ii. sec + i - sec i = 4iy x y2 2 6. If x + iy = cos u + iv then prove that b g 2 i. 1+ x y 2 7. Prove that tan -1 b g 2 2 b ii. 1- x y 2 cosh v - cosu LM tan 2 tanh 2 OP + tan LM tan tanh OP tan N tan 2 tanh 2 Q N tan tanh Q -1 b g 1 2 cot coth g u -1 sin x + iy then show that the argument of u is + where u +1 cosx sinh y cosx sinh y tan = , tan = 1+ sin x cosh y 1- sin x cosh y 8. If C.N. 26 Prove that 1. e sinh -1 x = log x + x2 + 1 j 2. L1 + 1 + x OP ( D'03) cosech x = log M MN x PQ 1 F 1 + x IJ ( J'02,N'95) tanh x = log G H1 - xK 2 2 3. 5. -1 -1 4. e L1 + x = log M MN cosh -1 x = log x + sech -1 6. coth -1 x = 1 2 j 1 - x O PP x Q F x + 1IJ log G H x - 1K x2 - 1 2 13 FE /Maths-I/ GQ/Complex number C.N. 27 Prove the following relations FG H 1. cosh -1 1+ x 2 = tanh -1 3. sech -1 sin = log cot 5. sinh -1 btan xg = log tan FGH 4 7. cosh -1 1 + x 2 = sinh -1 x IJ 1+ x K x 2 2 b g IJ K + x/2 b g b 2. tanh -1 sin = cosh -1 sec 4. tanh -1 x = sinh -1 b FG H tanh -1 cos = cosh -1 8. sinh -1 x = C.N. 28 x 1 - x2 IJ K bcosec g g 6. g 1 cosec -1 2 1 2x 1+ x 2 c h 1. Seperate the real and Imaginary parts of tan -1 e i c h 2. Prove that tan 1 e i c h 3. Prove that tan -1 e i LM FG N H n i log tan 2 4 2 4 2 LM FG N H n i log tan 2 4 2 4 2 IJ OP KQ IJ OP KQ 4. Prove that sin -1 e i cos1 sin i log c h sin 1 sin c h 1 sin sin 5. Prove that cos-1 e i sin 1 sin i log b 6. Seperate the real and imaginary parts of i. tan -1 x iy b g ii. tanh -1 x iy g C.N. 29 Prove that b g 1 log iz 1 z 2 i 3. cos1 ix = i log x + x 2 1 2 5. sinh -1 ix cosh 1 x i 2 e 1. sin -1 z = j b g b g 7. tan -1 z = e i log 2 e 2. sin -1 ix 2 n i log x 1 x 2 j e 4. cos-1 z = - i log z + z 2 1 b g 6. cosh -1 ix sinh 1 x i FG i + z IJ H i-z K b g 8. sin -1 cos ec j j 2 + i log cot 2 2 C.N. 30 Express in the form a + ib 3i i. cos-1 4 FG IJ H K C.N. 31 FG H 1 1+ i e i 1. If x + iy = log i 1 i e i 2. If cos FG iaIJ H4 K FG H bg ii. cos-1 i IJ K cosh b + i 3. Pr ove that tanh(log 5 ) then show that IJ 1 4K b g bg iii. sin -1 3i / 4 iv. sin -1 i x = and y = log sec + tan 2 b d then show that 2b = log 2 + 3 g i 2 3 14 b FE /Maths-I/ GQ/Complex number g 3. If tan x + iy i where x and y are real then prove that x is indeterminate and y is infinite. LM FG x a IJ OP N H x aKQ 4. Prove that tan -1 i i log a / x 2 b g C.N. 32 LOGARITHMS OF COMPLEX NUMBER Express in the form a + ib bg bg i. log -5 b g ii. logb -3g -2 iii. log 3 + 4i 4m +1 v. Prove that log i i = 4n +1 C.N. 33 LM sin bx + iyg OP 2i tan bcot x tanh yg N sin bx iyg Q F a + ib IJ 2i tan bb / ag log G H a - ib K L F a - ib IJ OP 2ab tan Mi log G N H a + ib K Q a b L F a + ib IJ OP a b cos Mi log G N H a - ib K Q a b L F a - ib IJ OP 2ab sin Mi log G N H a + ib K Q a b -1 1. Prove that log 2. Prove that 3. Prove that 4. Prove that 5. Prove that b g iv. log b1 - i g 1 + i -1 2 2 2 2 2 2 2 2 6. Prove that sin log(i -i ) 1 C.N. 34 1. 2. FG 1 IJ log FG 1 cosec IJ i FG IJ H 1- e K H 2 H 2 2K 2K 2 tan b b / a g y If ba + ibg m then prove that x log ca b h b g 1+ ig x b If i then prove that tan b / g y log2 b g 2 b1 ig Prove that log b1+ cos 2 i sin 2g log b2cos g i Prove that log p i 1 x iy 2 2 x iy 3. 4. -1 x iy C.N. 35 1. Seperate the real and imaginary parts by considering the principal values only. d 1-i 1 i i i) 1+ i iv) ib b g 1 iI F ii) G H 2 JK v) b1+ igb g d1i 3i 3 g 1 i iii) i i b gb vi) -i 1 i g C.N. 36 1. 2. 3. Find the principal value of i log b1 i g and show that its real part is e - 2 /8 FG log2IJ H4 K cos then prove that y = x tan tanq log x y b g d i Seperate d i i into real and imaginary parts by considering principal values only If log log x + iy p iq 2 2 i C.N. 37 1. If i +i i then prove that 2 2 eb4 n1g 15 i-- FG H FE /Maths-I/ GQ/Complex number IJ K A 2 B A A 2n + B 2 A iB, Prove that i) tan ii) A 2 + B2 = e- B 2. ii 3. If i i i B then prove that 4. Prove that i i cos i sin 5. Find the principal value of 6. Pr ove that the general value of (1 i tan ) i is e 2 m cos(log ) i sin(log ) 7. Find the principal value of 1+ i 3 A = b g where = 2n +1/ 2 e -b 2m+1/2 g i b1+ i tang d id i 1 i 3 i C.N. 38 1. If log sin x + iy = a + ib then prove that b g i) 2e 2a cosh 2y - cos 2x 2. Show that log tan b FG i x IJ i tan bsinh xg H 4 2K -1 g 3. If tan log x + iy a ib 4. b If 1+ i tan gb ii) tanb = cotx tanh y 1i tan c h then prove that tan log x 2 y 2 2a 1 a 2 b2 b g where a 2 b 2 1 g can have real value then show that it is sec esec j 2 ANSWERS Examples P. 2 1 3 i 2 2 5. 2 2i 6. i. z 7. i. F 1 2 I iFG 1IJ GH 2 JK H 2 K z 1 3 i 2 2 1 4 5. 0 . iii. z 2 x2 y2 x2 y2 x2 y2 1 ii. 1 iii. 1 5 9 25 16 4 5 DE MOIVRE’S THM 1. ii. z 1 2. cos FG 47 IJ H3 K + i sin FG 47 IJ H3 K 3. i 4 4. - 6. modulus = 4i , argument = / 6 C.N. 4 1. Modulus = cos sec LM OP N4 2 Q Argument = 4 2 C.N. 5 3. 2 16 C.N. 6 n n 1. 2 2 2 Cos 4 C.N. 9 2 2 4 2. 2 5 3 cos tan -1 3 3 3. cos 5 = cos5 10 cos3 sin 2 5 cos sin 4 sin 5 = 5 cos4 sin - 10 cos2 sin 3 + sin5 6. cos 6 = cos6 - 15 cos4 sin 2 + 15cos2 sin 4 - sin6 8. A = 6, B = 20, C = 6 16 FE /Maths-I/ GQ/Complex number C.N. 10 1 cos 6 - 6 cos 4 + 15 cos 2 -10 32 1 cos6 cos 6 + 6 cos 4 + 15 cos 2 + 10 32 1 6. cos8 cos 8 + 8 cos 6 28 cos 4 56 cos 2 35 128 1 8. cos5 Sin 3 sin 8 + 2 sin 6 - 2 sin 4 - 6 sin 2 128 2. sin 6 SUMMATION OF SERIES b 3. e x cos sin + x sin g C.N. 12 1. 2. 3. 4. 5. 6. 7. 8. 14. 15. 9. b g 12 + i sin b4k +1g 12 , k = 0,1, 2, 3, 4, 5 x = cos b4k +1g + i sin b4k +1g , k = 0, 1, 2 6 6 x = cos 4k +1 x = 2, 2i LM b g i sin b2k +1g OP, k = 0, 1, 2, 3, 4 5 5Q N x = cos b2k +1g + i sin b2k +1g , k = 0, 1, 2, 3, 4 and 5 5 F k I F k I x = cos G J + i sin G J k = 0, 1, 2, 3, H2K H2K x = cos b2k +1g + i sin b2k +1g , k = 0, 1, 2, 3, 4, 5 and 6 6 x = cos b2k +1g + i sin b2k +1g , k = 0, 1, 2 3 3 x = cos b2k +1g + i sin b2k +1g , k = 0, 1, 2, 3, 4, 5 and 6 6 O L x = 2 Mbcos 2k +1g + i sin b2k +1g P, k = 0, 1, 2 3 3Q N x = cos b4k +1g - i sin b4k +1g , k = 0, 1, 2, 3, and 8 8 x = cos b2k +1g + i sin b2k +1g , k = 0, 1, 2 3 3 x = 2 cos 2k +1 2k 2k + i sin , k = 1, 2, 3, 4, k 0 5 5 k k x = cos + i sin , k = 1, 2, 3, 4, 5 k 0 3 3 x = cos LM b g + i sin b4k +1g OP, k = 0, 1, 2, and 3 3Q N O L x = 2 Mcos b2k +1g + i sin b2k +1g P k = 0, 1, 2,3 4 4Q N x = cos b2k +1g + i sin b2k +1g , k = 0, 1, 2, 3, 4 and 5 5 O L x = b10g Mcos b2k +1g + i sin b2k +1g P k = 0, 1, 2, 3, 4 5 5Q N x = cos b2k +1g i sin b2k +1g , k = 0, 1, 3, 4, k 2 5 5 x = cos b2k +1g i sin b2k +1g , k = 0, 1, 2, 4, 5, 6, k 3 7 7 1 L O x = cosb2k +1g i sin b2k +1g P, k = 0, 1, 3, 4, k 2 M 2 N 5 5Q x = 4 cos 2k +1 3/2 10 1/ 5 11. 12. 13. 14. 15. 2k 2k + i sin , k = 1, 2, 3, 4, k 0 5 5 k k x = cos + i sin , k = 1, 2, 3, 4, 5 k 0 3 3 x = cos 17 LM N FE /Maths-I/ GQ/Complex number OP Q 1 2k 2k cos + i sin , k = 1, 2, 3, 4, k 0 2 5 5 1 x = , k = 0, 1, 2, 3, 4, 5, 6, 7 cos 2k +1 i sin 2k +1 -1 8 8 x = cos 2k +1 i sin 2k +1 , k = 0, 2, 3, 5 k 1,4 6 6 16. x = 17. b b 18. b b g g g g C.N. 14 FG 2k + / 3IJ , k 0,1,2 H 3 K x = cos b2k +1g i sin b2k +1g , k = 0, 1, 2, 3, 4, 5 and 6 6 2 21/3 cos 1. 3. k k i sin , k = 0, 1, 2, 3, 4, 5 3 3 k k common roots are x = + i sin , k = 1, 2, 4, 5 k 0,3 3 3 4 k -1 4k -1 1/ 3 2 sin / 2 cos i sin , k = 0, 1, 2 6 6 Z = cos 12k + 5 i sin 12k + 5 k = 0, 1, 2, 3, 4 30 30 x = cos 8. 9. g LMN b g b g b b b OP Q g g C.N. 15 1. 1- 2i, 1 3 i 2 2 1- i, 2 i 2. C.N. 16 1. Sin x cosh y + i cos x sinh y 3. sinh x cos y + i cosh x sin y sin 2x + i sinh 2y 5. cos 2x + cosh 2y C.N. 28 n i 1+ sin 1. log 2 4 4 1- sin 6. LM N L bx +1g 1 ii. log M 4 MN bx 1g i. cos x cosh y - i sin x sinh y 4. cosh x cos y + i sinh x sin y 6. sinh 2x + i sin 2y cos 2x + cos 2y IJ K OP + i log LM by +1g x Q 4 MN by 1g x L 2y OP y O i P tan M y PQ 2 N1 x y Q 1 2x tan -1 2 1 x2 y2 C.N. 30 1. i log 2 2 C.N. 32 1. FG H 2. a 2 2 2 2 2 OP PQ 2 -1 2 2 2 2. i log 2 d i 2 1 2 3. i log 2 4. d log 1+ 2 i f log 5 + i 2k + 1 a f a2n + 1f 2 i a2m 1f log 3 - a2n + 1f log 2 alog 3f2 a2n 1f2 2 1 log 2g i log 2 b 4 16 4 log 5 + i 2k + tan b4 / 3g 4. 1 log 2g b 4 16 log 2 log 3 + 2m + 1 2. 2 3. 2 -1 2 2 18 FE /Maths-I/ GQ/Complex number C.N. 35 LMcos FG 3 log 2IJ i sin FG 3 log 2IJ OP K H3 KQ N H3 IJ L K Cos FG log 2IJ i sin FG log 2IJ O MN H 4 K H 4 K PQ L F 1 I F 1 4. ie 5. 2 e Mcos G log 2J i sin G H4 2 N H4 2 K 1. 2e - / 2. FG 1 log 2 - /4 eH 2 3. 3 e - /2 /2 /4 IJ OP KQ log 2 6. e i /2 C.N. 36 3. e- /4 2 LMcos i sin OP 4 2Q N 42 C.N. 37 5. eb 2k + g b g b cos log cos - i sin log cos g 19
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