Write the complex number in polar form with argument 𝜃 between 0 and 2𝜋.
−7

2
 − 7

2
i

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.

Write the complex number in polar form with argument 𝜃 between 0 and 2𝜋: -7/2 - 7/2i

Learn how to convert the complex number -7/2 - 7/2i into its polar form, with the magnitude and argument θ between 0 and 2π. This detailed solution includes step-by-step calculations for finding the magnitude and determining the correct angle using trigonometry. Ideal for students in Grades 11-12 studying complex numbers and polar coordinates.

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