QUESTION 9

9.1State De Moivre's Theorem.

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9.2Express cos 5 and sin 4 as polynomials in terms of sin and cos .

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9.3Let w be a negative real number, za 6th root of w.

(a)Show that z (k) = 6

6th roots of w.

cos

( +2k )

+ i sin

( +2k )

, k= 0,1,2,3,4,5 is a formula for the

Show all your working.

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(b)Hence determine the 6th roots of 729.

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(c)Given z = cos + i sin and u + iv = (1 + z)(1 + z2). Prove that v = u tan( 3 ) and

u2 + v2 = 16 cos2( ) cos2()

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