Math Problem Statement

z0 ≈ 0.6687∠(−0.0899) [or 0.6660 − 0.0601j] z1 ≈ 0.6687∠3.0517 [or − 0.6660 + 0.0601j] These are the answers

Solution

You seem to be presenting the results for two complex numbers in both polar and rectangular forms.

Given:

  • z00.6687(0.0899)z_0 \approx 0.6687 \angle (-0.0899) or z00.66600.0601jz_0 \approx 0.6660 - 0.0601j
  • z10.66873.0517z_1 \approx 0.6687 \angle 3.0517 or z10.6660+0.0601jz_1 \approx -0.6660 + 0.0601j

Let's break it down:

  1. Rectangular Form: The complex number is given as a+bja + bj, where aa is the real part and bb is the imaginary part.

    • z0=0.66600.0601jz_0 = 0.6660 - 0.0601j
    • z1=0.6660+0.0601jz_1 = -0.6660 + 0.0601j
  2. Polar Form: A complex number can also be written as rθr \angle \theta, where rr is the modulus (magnitude) and θ\theta is the argument (angle in radians).

    • z00.66870.0899z_0 \approx 0.6687 \angle -0.0899
    • z10.66873.0517z_1 \approx 0.6687 \angle 3.0517

Key Observations:

  • The magnitudes rr are approximately the same: z0z10.6687|z_0| \approx |z_1| \approx 0.6687.
  • The angles (arguments) are almost opposite (one is slightly negative, and the other is close to π\pi).

Do you need further explanation on converting between polar and rectangular forms, or is there a specific question about these results?

Here are 5 follow-up questions to deepen understanding:

  1. How do you convert a complex number from polar to rectangular form?
  2. How do you find the argument of a complex number in radians?
  3. What is the geometric interpretation of these two complex numbers on the complex plane?
  4. How do you multiply two complex numbers in polar form?
  5. How can the polar form be useful in solving AC circuit problems?

Tip: In polar form, the multiplication of complex numbers is simpler since you multiply their magnitudes and add their angles.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Complex Numbers
Polar and Rectangular Forms

Formulas

Rectangular Form: z = a + bj
Polar Form: z = r∠θ where r = |z| and θ = arg(z)

Theorems

Conversion between Polar and Rectangular Forms
Argument of a Complex Number

Suitable Grade Level

University-Level Mathematics or Advanced High School (Grades 11-12)