Let z=(cosθ+i sinθ)/(cosθ-i sinθ), (π)/(4)<θ<(π)/(2). Then (z) is (a) 2 θ(b) 2 θ-π(c) π+2 θ(d) 2

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Let z=(cosθ+i sinθ)/(cosθ-i sinθ), (π)/(4)<θ<(π)/(2). Then...

Let $\mathrm{z}=\frac{\cos \theta+\mathrm{i} \sin \theta}{\cos \theta-\mathrm{i} \sin \theta}, \frac{\pi}{4}<\theta<\frac{\pi}{2}$. Then $\arg (\mathrm{z})$ is
(a) $2 \theta$
(b) $2 \theta-\pi$
(c) $\pi+2 \theta$
(d) $2 \theta-2 \pi$

Key Concepts

Argument of a Complex Number

The argument of a complex number is the angle made with the positive real axis in the complex plane. It represents the direction of the vector corresponding to the complex number. Correctly determining the argument, especially in operations such as division, is crucial for understanding the geometric interpretation of complex numbers.

Division of Complex Numbers in Polar Form

When complex numbers are expressed in polar form, division becomes straightforward since the magnitudes divide and the arguments subtract. Specifically, if two numbers have arguments ?1 and ?2, then the argument of their quotient is given by ?1 - ?2. This property simplifies many problems involving the manipulation of complex fractions.

Euler's Formula

Euler's Formula, which states that e^(i?) = cos? + i sin?, provides a powerful connection between complex exponentials and trigonometric functions. This formula allows complex numbers to be expressed in a concise form and is fundamental in analyzing rotations and oscillations in the complex plane.

Polar Representation of Complex Numbers

The polar representation expresses a complex number in terms of its magnitude and angle (or argument), typically written as r·e^(i?). This form is especially useful for multiplication, division, and finding powers or roots of complex numbers because these operations reduce to simple arithmetic on the magnitudes and arguments.

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