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MATH 1111-Tutorial VI-Complex Numbers

1. Find the two square roots of the complex number w = 8 + 6i. Using the above result, find the two roots a and b of the equation 4z 2 − 2(1 + 3i)z − 4 = 0. Let a be the root for which Re a > 0. Show that a b 4

= 16.

√ 2. Find the modulus and argument of z = −1 + i 3 and w = 1 − i. Hence show that z3 a= 4 w is purely real. 3. Use De Moivre’s Theorem to express sin 4θ and cos 4θ in terms of sin θ and cos θ. Hence show that (i) cos 4θ = 8 cos4 θ − 8 cos2 θ + 1. (ii) tan 4θ = 4. Let z = eiθ . By expanding 1 z− z show that sin5 θ = 1 (sin 5θ − 5 sin 3θ + 10 sin θ) . 16 1 5

4 tan θ − 4 tan3 θ . 1 − 6 tan2 θ + tan4 θ

,

Hence show that
0

π 2

sin5 θ dθ =

8 . 15

5. If z = eiθ , show that for k ∈ Z, zk + Hence show that cos8 θ = 1 (cos 8θ + 8 cos 6θ + 28 cos 4θ + 56 cos 2θ + 35) . 128 1 = 2 cos kθ. zk

Using the above result, show that
π 3

cos8 θ dθ =

0

√ 1 560π + 507 3 . 6144

6. (a) Find all the roots of the equation z 6 + 64 = 0. (b) If w = eiθ , show that w−1 θ = i tan . w+1 2 (c) Show that the six roots of the equation (1 + w)6 + 64(1 − w)6 = 0 are given by wk = ±i tan (2k + 1)π , k = 0, 1, 2. 12

2