APPLYING DE MOIVRES THEOREM PRACTICE PROBLEMS

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

[(a + b ω + cω²)/(b + c ω + a ω²)] + [(a + b ω + cω²)/(c + a ω + b ω²)]  =  -1

(2)  Show that 

(3)  Find the value of 

(4)  If 2 cos α  =  x + (1/x) and 2 cos β  =  y + (1/y), show that 

(i)  (x/y) + (y/x)  =  2 cos (α - β)

(ii)   xy - (1/xy)  =  2i sin (α + β)

(iii)  (x^m/yⁿ) -  (yⁿ/x^m)  =  2i sin (mα - nβ)

(iv)  (x^myⁿ) +  1/(x^myⁿ)   =  2 cos (mα + nβ)

Solution

(5)  Solve the equation z³ + 27 = 0.        Solution

(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

Solution

(7)  Find the value of

Solution

(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^{2^11})  =  1

Solution

(9)  If z = 2 - 2i, find rotation of z by θ radians in the counter clock wise direction about the origin when

(i)  θ  =  π/3    (ii) θ  =  2π/3    (iii) θ  =  3π/2

Solution 

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