The image contains steps for converting a complex number $8 + 6j$ from rectangular form to polar form.

Breakdown of Steps in the Image:

1. Rectangular Form: $z = x + iy$ Here, $x = 8$ and $y = 6$.

2. Magnitude ($r$): $r = \sqrt{x^2 + y^2}$ Substituting values: $r = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10$

3. Argument ($\theta$): $\theta = \tan^{-1}\left(\frac{y}{x}\right)$ Substituting values: $\theta = \tan^{-1}\left(\frac{6}{8}\right)$ Simplifying: $\theta = \tan^{-1}(0.75) \approx 36.8^\circ \approx 37^\circ$

4. Polar Form: The polar form of $z$ is: $z = r(\cos \theta + i\sin \theta)$ Substituting $r = 10$ and $\theta = 37^\circ$: $z = 10(\cos 37^\circ + i\sin 37^\circ)$

Would you like further clarification on any part?

5 Related Questions:

1. How do you convert polar form back to rectangular form?
2. What is the significance of the magnitude and argument in polar representation?
3. How do you compute $\tan^{-1}$ for angles in radians?
4. How is this process different if $z$ lies in a different quadrant?
5. Can you explain the relationship between polar and exponential forms of complex numbers?

Tip:

Always ensure that your calculator is set to the correct mode (degrees or radians) when finding arguments of complex numbers.