To convert the complex number $-7 + 3i$ to polar form, we follow these steps:

1. Find the modulus $r$: $r = \sqrt{(-7)^2 + 3^2} = \sqrt{49 + 9} = \sqrt{58}$

2. Calculate the argument $\theta$: The argument $\theta$ is given by: $\theta = \tan^{-1} \left(\frac{3}{-7}\right)$ Since the complex number is in the second quadrant (negative real part and positive imaginary part), we add $\pi$ to the angle obtained.

   $\theta = \tan^{-1}\left(\frac{3}{-7}\right) + \pi$

   Using an approximation: $\theta \approx -0.404 + \pi \approx 2.738 \text{ radians}$

3. Express in polar form: The polar form is: $r(\cos \theta + i \sin \theta)$ or equivalently, $\sqrt{58} \left(\cos(2.738) + i \sin(2.738)\right)$

Thus, the polar form of $-7 + 3i$ is: $\sqrt{58} \operatorname{cis}(2.738)$

Would you like further details or have any questions?

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Here are some related questions for further exploration:

1. What is the significance of the modulus and argument in polar coordinates?
2. How would you convert from polar form back to rectangular form?
3. What are some applications of polar form in physics or engineering?
4. How would polar coordinates change if the complex number were in the third quadrant?
5. How do you determine the quadrant based on real and imaginary parts?

Tip: When converting to polar form, always check the quadrant to ensure the correct angle.