Exercise 2.8: de Moivre’s Theorem and its Applications

EXERCISE 2.8

1. If ω ≠ 1 is a cube root of unity, show that 

2. Show that 

3. Find the value of  

4. If 2cosα = x + [1/x] and 2cos β = y + [1/y] , show that

5. Solve the equation z3 + 27 = 0 .

6. If ω ≠ 1 is a cube root of unity, show that the roots of the equation ( z -1)3 + 8 = 0 are -1, 1- 2ω, 1- 2ω2 .

7. Find the value of 

8. If ω ≠ 1 is a cube root of unity, show that 

(i) (1- ω + ω2)6 + (1+ ω - ω2 )6 = 128.

(ii) (1+ ω )(1+ ω2)(1+ ω4 )(1+ ω8 )... ...(1+ ω 2.pow(11) ) = 1.

9. If z = 2 - 2i , find the rotation of z by θ radians in the counter clockwise direction about the origin when

(i) θ = π/3

(ii) θ = 2π/3

(iii) θ = 3π ./2

10. Prove that the values of 4√-1 are ± 1/√2 (1±i).

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