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

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

$r(\cos \theta + i \sin \theta)$

Where:

• $r$ is the modulus of the complex number.
• $\theta$ is the argument (angle) of the complex number.

Step 1: Find the modulus $r$

The modulus $r$ is given by:

$r = \sqrt{a^2 + b^2}$

where $a$ is the real part and $b$ is the imaginary part of the complex number $z = a + bi$.

For $5 - 5i$:

• $a = 5$
• $b = -5$

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

Step 2: Find the argument $\theta$

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

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

For $5 - 5i$: $\theta = \tan^{-1}\left(\frac{-5}{5}\right) = \tan^{-1}(-1)$

The value of $\tan^{-1}(-1)$ is $315^\circ$ (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:

$5 - 5i = 5\sqrt{2} \left( \cos 315^\circ + i \sin 315^\circ \right)$

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 $315^\circ$ in this case and not $45^\circ$?
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.