Applying De Moivres Theorem Practice Problems :
Here we are going to see some practice problems on 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) (xm/yn) - (yn/xm) = 2i sin (mα - nβ)
(iv) (xmyn) + 1/(xmyn) = 2 cos (mα + nβ)
(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
(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 + ω2^11) = 1
(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
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