DE MOIVRE'S THEOREM AND ITS APPLICATIONS

Theorem :

For any complex number cos θ + i sin θ and any integer n,

(cos θ + i sin θ)n = cos nθ + i sin nθ

Corollary :

(cos θ - i sin θ)n = cos nθ - i sin nθ

(cos θ + i sin θ)-n = cos nθ - i sin nθ

(cos θ - i sin θ)-n = cos nθ + i sin nθ

Solved Problems

Problem 1 :

Simplify :

Solution :

Problem 2 :

Simplify :

Solution :

Problem 3 :

If z = cos θ + i sin θ, find

Solution :

Part (i) :

Part (ii) :

Problem 4 :

Simplify :

Solution :

Let z = cos 2θ + i sin 2θ. Then


For any complex number z, if |z| = 1, then


Problem 5 :

Simplify :

Solution :

Problem 6 :

Simplify :

(1 + i)18

Solution :

Write the givcen complex number in polar form.

(1 + i)18 = r(cos θ + i sin θ)

The point on the Argand plane corresponding to the complex number (1 + i) is (1, 1).

And the point (1, 1) lies in the first quadrant.

In the first quadrant,

Writing the given complex number in polar form,

Raising to the power of 18 on both sides,

Problem 7 :

Simplify :

(-√3 + 3i)31

Solution :

Write the givcen complex number in polar form.

(-√3 + 3i)31 = r(cos θ + i sin θ)

The point on the Argand plane corresponding to the complex number (-√3 + 3i) is (-√3, 3).

And the point (-√3, 3) lies in the second quadrant.

In the second quadrant,

Writing the given complex number in polar form,

Raising to the power of 31 on both sides,

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