# 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.

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.

Writing the given complex number in polar form,

Raising to the power of 31 on both sides,

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