91Ó°ÊÓ

Complex numbers can be represented in two different forms: polar and rectangular. In the polar form, a complex number is expressed in terms of its magnitude and angle. The given exercise provides a complex number in the form \(z = r \operatorname{cis}(\theta)\), where \(r\) is the magnitude, and \(\theta\) is the angle with the positive x-axis, measured in radians.

To convert this number into its rectangular form, we utilize the expression \(z = r (\cos(\theta) + i\sin(\theta))\). The rectangular form involves real and imaginary parts, expressed as \(x + yi\), where \(x\) is the real part and \(yi\) is the imaginary part.

• Given: \(r = 4\) and \(\theta = \frac{2\pi}{3}\).
• Rectangular form is obtained by multiplying \(r\) with both the cosine and sine of the angle.

This transformation highlights the seamless conversion between polar coordinates \((r, \theta)\) and rectangular coordinates \((x, y)\).

The CIS identity is a handy tool in complex numbers. It offers a simplified way of expressing trigonometric expressions. Specifically, \(\operatorname{cis}(\theta)\) denotes \(\cos(\theta) + i\sin(\theta)\). This reveals that a polar complex number can be rapidly decomposed into its rectangular components.

In our exercise, the expression \(z = 4 \operatorname{cis}\left(\frac{2\pi}{3}\right)\) becomes \(z = 4 (\cos\left(\frac{2\pi}{3}\right) + i\sin\left(\frac{2\pi}{3}\right))\). Using the CIS identity simplifies the transition from polar to the familiar rectangular form, which is crucial for calculation and visualization.

• Saves time with complex arithmetic.
• Enhances understanding of angle-related behavior of complex numbers.

A key aspect of converting complex numbers is understanding where the angle \(\theta\) lies in the polar coordinate system. The standard unit circle is divided into four quadrants. Each quadrant affects the sign of sine and cosine values differently, which is crucial in determining the rectangular coordinates.

For \(\theta = \frac{2\pi}{3}\), the angle is situated in the second quadrant. In this quadrant:

• \(\cos\left(\frac{2\pi}{3}\right)\) is negative because cosine values are negative in the second quadrant.
• \(\sin\left(\frac{2\pi}{3}\right)\) remains positive, matching the trend for sine in this quadrant.

Understanding these fundamental trigonometric characteristics ensures accurate conversion from polar to rectangular form.

Converting between polar and rectangular forms involves understanding the relationship between angles and magnitudes. For instance, if you know the rectangular form, you can compute the polar form using the formulas:
- Magnitude \(r = \sqrt{x^2 + y^2}\)
- Angle \(\theta = \arctan\left(\frac{y}{x}\right)\) (considering the quadrant of the angle for the correct calculation)

In the original exercise, the focus is on converting from polar to rectangular, which requires clear comprehension of trigonometric functions and their interaction within quadrants. The capability to fluidly switch between these forms enhances problem-solving versatility in complex number mathematics. Such transformations simplify comparisons and algebraic operations, giving you a more profound understanding of the mathematical landscape.