91Ó°ÊÓ

In this exercise, we start by focusing on the process of dividing the magnitudes when dealing with complex numbers in polar form. The magnitude of a complex number - represents its "distance" from the origin in the complex plane. For the given numbers, we have:

• Magnitude of \( z_1 = \sqrt{40} \)
• Magnitude of \( z_2 = \sqrt{10} \)

To find the quotient's magnitude, divide the magnitudes of \( z_1 \) and \( z_2 \): \[ \frac{\sqrt{40}}{\sqrt{10}} = \sqrt{4} = 2 \] The result is \( 2 \), which is the magnitude of the new complex number after division. This step is critical as it simplifies our complex number to a more manageable form.

Once our complex number is expressed in polar form, we aim to convert it into a rectangular form for practical computation. This involves knowing the relationship between the trigonometric functions in polar coordinates and the standard form \( a + bi \) of complex numbers. Given the polar form, \( 2(\cos 90^{\circ} + i\sin 90^{\circ}) \), we transition to rectangular form by calculating:

• \( \cos 90^{\circ} = 0 \)
• \( \sin 90^{\circ} = 1 \)

This means: \( 2(0 + i\cdot1) = 2i \). In rectangular terms, \( 2i \) suggests there is no "real" part, and the entire magnitude is purely imaginary. This form is often preferred for final answers in complex arithmetic, making solving problems involving addition or subtraction more straightforward.

Subtracting angles is a central operation when dividing complex numbers in polar form. It ensures the direction of the resulting complex number is correct. The angles, interpreted as "directions," define the positioning within the complex plane. With our numbers:

• \( z_1 \) has an angle of \( 110^{\circ} \)
• \( z_2 \) has an angle of \( 20^{\circ} \)

By subtracting these: \( 110^{\circ} - 20^{\circ} = 90^{\circ} \). This angle calculation is crucial as it dictates where on the complex plane our result, \( 2(\cos 90^{\circ} + i\sin 90^{\circ}) \), will lie. A subtraction operation ensures that we correctly orient the magnitude we calculated previously.

Complex numbers are often expressed in polar form to better handle multiplication and division. Polar form combines a magnitude and an angle, offering a geometric perspective of complex numbers. In this case: \( z_1 = \sqrt{40}(\cos 110^{\circ} + i \sin 110^{\circ}) \) and \( z_2 = \sqrt{10}(\cos 20^{\circ} + i \sin 20^{\circ}) \). When divided, their magnitudes and angles are separately processed to give the quotient: \[ 2(\cos 90^{\circ} + i\sin 90^{\circ}) \]. Polar representation is especially helpful because:

• It simplifies complex division into manageable arithmetic (dividing magnitudes and subtracting angles).
• Makes visualization in the complex plane straightforward.
• Supports easy conversion back to rectangular form for final answers.

Understanding polar form is essential for anyone tackling complex number operations, serving as a bridge between geometric interpretation and algebraic computation.