Math Problem Statement

Write the complex number in polar form with argument θ between 0 and 2π: 5 - 5i

Solution

To write the complex number 55i5 - 5i in polar form with argument θ\theta between 00 and 2π2\pi, follow these steps:

Step 1: Calculate the modulus rr

The modulus rr is given by: r=x2+y2r = \sqrt{x^2 + y^2} where x=5x = 5 and y=5y = -5.

r=52+(5)2=25+25=50=52r = \sqrt{5^2 + (-5)^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2}

Step 2: Calculate the argument θ\theta

The argument θ\theta is given by: θ=tan1(yx)\theta = \tan^{-1}\left(\frac{y}{x}\right) θ=tan1(55)=tan1(1)\theta = \tan^{-1}\left(\frac{-5}{5}\right) = \tan^{-1}(-1)

The arctangent of 1-1 in the coordinate system places the angle in the fourth quadrant. Thus: \theta = \frac{7\pi}{4} \, \text{(since \tan^{-1}(-1)correspondstocorresponds to-\frac{\pi}{4},butweadjusttobeintherange, but we adjust to be in the range [0, 2\pi])}

Step 3: Write the polar form

The polar form of a complex number is: z=r(cosθ+isinθ)z = r (\cos\theta + i\sin\theta)

Substituting the values r=52r = 5\sqrt{2} and θ=7π4\theta = \frac{7\pi}{4}: z=52(cos7π4+isin7π4)z = 5\sqrt{2} \left(\cos\frac{7\pi}{4} + i\sin\frac{7\pi}{4}\right)

Alternatively, using Euler's formula, the polar form can also be expressed as: z=52ei7π4z = 5\sqrt{2} e^{i\frac{7\pi}{4}}

Let me know if you'd like any additional clarifications!


Related Questions:

  1. How can the polar form of a complex number be converted back to rectangular form?
  2. Why is the argument θ\theta adjusted to the range [0,2π][0, 2\pi]?
  3. What is the significance of Euler's formula in the context of polar coordinates?
  4. How do the quadrant positions of a complex number affect the sign of the argument?
  5. What are the practical applications of representing complex numbers in polar form?

Tip:

Always remember that in polar form, the angle θ\theta depends on the quadrant of the complex number. Use the signs of xx and yy to correctly determine its placement.

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Math Problem Analysis

Mathematical Concepts

Complex Numbers
Polar Coordinates
Trigonometry

Formulas

r = √(x² + y²)
θ = tan⁻¹(y / x)
z = r(cosθ + i sinθ)
z = re^(iθ)

Theorems

Euler's Formula

Suitable Grade Level

Grades 11-12