If x+i y=(1-i √(3))^160 then (x, y) is (a) (2^99, 2^99 √(3)) (b) (2^100, 2 √(3)) (c) (-2^99, 2^9

step 1 So in the given question we are told that x plus iy is equal to 1 minus i root 3 raised to the p

If x+i y=(1-i √(3))^160 then (x, y) is (a) (2^99, 2^99 √(3))...

If $x+i y=(1-i \sqrt{3})^{160}$ then $(x, y)$ is
(a) $\left(2^{99}, 2^{99} \sqrt{3}\right)$
(b) $\left(2^{100}, 2 \sqrt{3}\right)$
(c) $\left(-2^{99}, 2^{99} \sqrt{3}\right)$
(d) $\left(-2^{100}, 2 \sqrt{3}\right)$

Key Concepts

Angle Reduction (Modulo 2? Periodicity)

When performing operations on complex numbers in polar form, the angle (or argument) can be reduced modulo 2? since the trigonometric functions are periodic. This reduction allows one to simplify the computed angle to an equivalent angle within the standard interval, thereby making the final result easier to interpret.

De Moivre’s Theorem

De Moivre’s theorem establishes that raising a complex number in polar form to an integer power is accomplished by raising the modulus to that power and multiplying the angle by the exponent, i.e., [r(cos ? + i sin ?)]^n = r^n (cos(n?) + i sin(n?)). This theorem is fundamental in simplifying the computation of complex number powers.

Euler’s Formula

Euler’s formula states that e^(i?) is equal to cos ? + i sin ?, providing a powerful connection between complex exponentials and trigonometric functions. This relationship is essential when converting complex numbers to and from polar form.

Complex Numbers in Polar Form

Expressing a complex number in polar form involves writing it as a product of a modulus (its absolute value) and a complex exponential (or equivalently, using cosine and sine), i.e. r(cos ? + i sin ?). This representation simplifies operations such as exponentiation and multiplication.

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