Math Problem Statement

Write the complex number 5 - 5i in trigonometric form r( cos θ + i sin θ), with θ in the interval [0°,360°).

Solution

Let's express the complex number 55i5 - 5i in trigonometric form.

The general form of a complex number in trigonometric (polar) form is:

r(cosθ+isinθ)r(\cos \theta + i \sin \theta)

Where:

  • rr is the modulus of the complex number.
  • θ\theta is the argument (angle) of the complex number.

Step 1: Find the modulus rr

The modulus rr is given by:

r=a2+b2r = \sqrt{a^2 + b^2}

where aa is the real part and bb is the imaginary part of the complex number z=a+biz = a + bi.

For 55i5 - 5i:

  • a=5a = 5
  • b=5b = -5

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

Step 2: Find the argument θ\theta

The argument θ\theta is the angle that the complex number makes with the positive real axis, which can be calculated as:

θ=tan1(ba)\theta = \tan^{-1}\left(\frac{b}{a}\right)

For 55i5 - 5i: θ=tan1(55)=tan1(1)\theta = \tan^{-1}\left(\frac{-5}{5}\right) = \tan^{-1}(-1)

The value of tan1(1)\tan^{-1}(-1) is 315315^\circ (since the complex number is in the fourth quadrant, where both real part is positive and imaginary part is negative).

Final Trigonometric Form

Now, we can write the complex number in trigonometric form:

55i=52(cos315+isin315)5 - 5i = 5\sqrt{2} \left( \cos 315^\circ + i \sin 315^\circ \right)

Let me know if you need further clarification or a breakdown of any specific steps.


Relative Questions:

  1. How do you find the modulus of a complex number?
  2. What is the argument of a complex number and how is it related to the real and imaginary parts?
  3. How do you convert a complex number from rectangular to polar form?
  4. Why is the argument 315315^\circ in this case and not 4545^\circ?
  5. What are the uses of trigonometric form in complex number calculations?

Tip: Always check the quadrant of the complex number when determining the argument to ensure the angle is correct.

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Math Problem Analysis

Mathematical Concepts

Complex Numbers
Trigonometry
Polar Form

Formulas

Modulus formula: r = √(a^2 + b^2)
Argument formula: θ = tan^{-1}(b/a)

Theorems

Polar form of a complex number

Suitable Grade Level

Grades 10-12