To write the complex number $-\frac{7}{2} - \frac{7}{2}i$ in polar form, we need to determine its magnitude $r$ and argument $\theta$.

Step 1: Calculate the Magnitude

The magnitude $r$ of a complex number $a + bi$ is given by: $r = \sqrt{a^2 + b^2}$ For the complex number $-\frac{7}{2} - \frac{7}{2}i$:

$a = -\frac{7}{2}, \quad b = -\frac{7}{2}$

$r = \sqrt{\left(-\frac{7}{2}\right)^2 + \left(-\frac{7}{2}\right)^2}$

$r = \sqrt{\left(\frac{49}{4}\right) + \left(\frac{49}{4}\right)}$

$r = \sqrt{\frac{98}{4}}$

$r = \sqrt{24.5}$

$r = \frac{7}{\sqrt{2}}$

$r = \frac{7\sqrt{2}}{2}$

Step 2: Calculate the Argument

The argument $\theta$ of a complex number $a + bi$ is given by: $\theta = \tan^{-1}\left(\frac{b}{a}\right)$

Since both $a$ and $b$ are negative, the complex number lies in the third quadrant. For $-\frac{7}{2} - \frac{7}{2}i$:

$\theta = \tan^{-1}\left(\frac{-\frac{7}{2}}{-\frac{7}{2}}\right)$

$\theta = \tan^{-1}(1)$

$\theta = \frac{3\pi}{4} \text{ (since it's in the third quadrant)}$

Step 3: Adjust for the Third Quadrant

In the third quadrant, the argument is:

$\theta = \pi + \frac{\pi}{4} = \frac{5\pi}{4}$

Conclusion: Polar Form

Thus, the polar form of the complex number $-\frac{7}{2} - \frac{7}{2}i$ is: $\frac{7\sqrt{2}}{2} \left(\cos\left(\frac{5\pi}{4}\right) + i\sin\left(\frac{5\pi}{4}\right)\right)$

Summary

The polar form is: $\boxed{\frac{7\sqrt{2}}{2} \text{cis} \left( \frac{5\pi}{4} \right)}$

Do you want more details or have any questions?

Relative Questions:

1. How do you convert polar coordinates back to rectangular form?
2. What is the significance of the argument in the polar form of a complex number?
3. How can you determine the quadrant of a complex number?
4. What is Euler's formula and how does it relate to polar form?
5. Can you explain how to add two complex numbers in polar form?

Tip:

When working with complex numbers in polar form, always make sure to identify the correct quadrant for the argument to avoid mistakes in determining the angle.