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De Moivre’s Theorem — Statement, i) If ‘n’ is an integer (positive or negative) then, (cos 6+ i sin 8)" = cos n@ +i sin nd, ii) If‘n’ isa fraction, then cos nO + i sin n@ is one of the values of (cos 6+ isin 6)", Example:, i) (osO+isin®)” = cos 30+isin 30, ii) (cos 6+ i sin 0)! = cos (-0) +i sin (-0), , = cos @-isin®, , Results:, 1) (cos@+isin@)" = cos (-nd)+isin (-n6), = cos né-i sin nd, , 1 as er, 2)! Sepa = (cos 0+ i sin 6), , = cos (-0) +i sin (-8), = cos 6-isin®, , 1 _ des ail, 3) Seria = (cos 0-i sin 6), = cos (-0)-i sin (-0), , = cos +i sin®, , 4) sin@+icos@ = cos (90-0) +i sin (90-0), , = cos (F-0)+i sin ($0), , , , , , MATH FACT, (i) sin @+icos@= i (cos 0 —i sin), (i) sin@—icos®@ = -i(cos0+isin 0), , (iii) cos (-0) = cos @, (iv) sin (0) =-sin 0, , , , , , Note :, 1. Ifa=cos a+isina, b=cos +i sin B then, (i) ab = (cos a+ i sin a) (cos B + i sin B), , ab = cos (a+8) +i sin (a+), , cosatisina, cosp+ising, , , , il, , , , a, b, 3 cos (a- B) + i sin (a- B), , , , , , , , 2. Ifa =cosa+ isin a, b=cosB +isin Band c=cosy+isiny then, , abc = (cosa+i sin a)(cosB+i sin B) (cos y+i sin y), , abe = cos(a-+B+y) + i sin (a+B+Y)

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20) Simplify, , Solution :, , Given, , 21) Simplify, , Solution :, , Given, , 22) Simplify, , Solution :, , Given, , (cos 30 + i sin 30)*(cos 46 +i sin 40)?, “(cos 26 +i sin 20)5(cos SO + i sin 50)?, , , , (cos 30 +i, (cos 20 +i, , in 30)*(cos 40 + isin 40)?, in 20)5 (cos 50 + isin 50)3, , , , , (cos 6 + i sin 0)*3(cos 0 + isin 0)?**, (Cos 6 + i sin 0)5%2 (cos 0 + i sin 8)?*5, , (cos 0 +i sin 6)?2(cos 6 +isin6)®, (cos 6 + i sin 8)!°(cos 0 +i sin 0)*5, = (cos 0 + isin @)12*+8-10-15, , = (cos@+isin6)~>, , = cos5@—isin50, , (cos 20~ i sin 20)* (cos 40 + isin 40)~5, “(cos 30 + i sin 36)?(cos 50—-i sin 58)~3, , (cos 20— i sin 20)*(cos 40 +i sin 40)~5, (cos 30 + i sin 30)2(cos 50- i sin 50)~3, , _ (Cos +i sin 8)**-2(cos 0 + isin 0)-5**, ~ (cos @ +i sin 6)?*3(cos 0 + isin 0)~3x-5, , _ (cos 0 +i sin 0)~8(cos 0 + isin 0)~2°, (cos 0 +i sin 6)6(cos 6 +i sin 6)!5, , (cos @ +i sin 0)~8-20-6-15, , (cos © + isin 6)~*9, , cos 496 — isin 498, , _(cos 30 + i sin 36)~5(cos 20 + i sin 26)*, * (cos 40— i sin 40)-2(cos 50— i sin 50)3, , (cos 30 +i sin 30)-*(cos 20 +i sin 20)*, (cos 40— i sin 46)~?(cos 50 i sin 50)3, , (cos 6 + isin )~5*3(cos 0 + i sin 0)**?, ~~ (cos @ + i sin 6)-2%-4(cos 6 + i sin 8)3*-5, , _ (cos + isin )~*5(cos 6 + isin 6)®, ~~ (cos 0 + isin 0)8(cos 6 + isin 6)-*5, , (cos 0 + i sin @)~15+8-8415, , (cos 6 + isin 0)°, , cos0 +isin0, , 1+i(o) [+ cos 0=1, sin 0 = 0], =