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)18Solution :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)31Solution :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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