Math Problem Statement

Express Z=-2+2i3^1/2

Solution

To express the complex number Z=2+2i3Z = -2 + 2i\sqrt{3} in polar form, we need to find its magnitude rr and angle θ\theta.

Step 1: Calculate the magnitude rr

The magnitude rr of a complex number Z=a+biZ = a + bi is given by: r=a2+b2r = \sqrt{a^2 + b^2} For Z=2+2i3Z = -2 + 2i\sqrt{3}, where a=2a = -2 and b=23b = 2\sqrt{3}: r=(2)2+(23)2=4+12=16=4r = \sqrt{(-2)^2 + (2\sqrt{3})^2} = \sqrt{4 + 12} = \sqrt{16} = 4

Step 2: Calculate the angle θ\theta

The angle θ\theta (also known as the argument) is found using: θ=tan1(ba)\theta = \tan^{-1}\left(\frac{b}{a}\right) For Z=2+2i3Z = -2 + 2i\sqrt{3}: θ=tan1(232)=tan1(3)\theta = \tan^{-1}\left(\frac{2\sqrt{3}}{-2}\right) = \tan^{-1}\left(-\sqrt{3}\right) Since the real part a=2a = -2 is negative and the imaginary part b=23b = 2\sqrt{3} is positive, the complex number is in the second quadrant. Therefore, we need to add 180180^\circ to the principal angle found: θ=180+tan1(3)=18060=120\theta = 180^\circ + \tan^{-1}\left(-\sqrt{3}\right) = 180^\circ - 60^\circ = 120^\circ

Step 3: Write the polar form

Thus, the polar form of Z=2+2i3Z = -2 + 2i\sqrt{3} is: Z=[4,120]Z = [4, 120^\circ]

Final Answer:

Z=4(cos120+isin120)Z = 4\left(\cos 120^\circ + i\sin 120^\circ\right) or simply, Z=[4,120]Z = [4, 120^\circ]

Would you like more details or have any questions?

Here are five related questions:

  1. How would you convert a complex number from polar form back to rectangular form?
  2. What is the significance of the argument in the polar form of a complex number?
  3. How do you find the polar form of a complex number if it lies in the third or fourth quadrant?
  4. What is the geometric interpretation of the magnitude and angle in polar form?
  5. How would the polar form change if the complex number was multiplied by a scalar?

Tip: When determining the angle in polar form, always consider the correct quadrant of the complex number to get the proper direction of the angle.

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Math Problem Analysis

Mathematical Concepts

Complex Numbers
Polar Form
Trigonometry

Formulas

Magnitude of a Complex Number
Angle of a Complex Number in Polar Form

Theorems

Argument (Angle) of a Complex Number

Suitable Grade Level

Advanced High School