The problem is to convert the complex number $10 \cdot e^{j60^\circ}$ into its rectangular and polar forms.

Solution:

1. Polar Form:
   Given $10 \cdot e^{j60^\circ}$, this is already in polar form, where:

   • The magnitude $r = 10$.
   • The angle $\theta = 60^\circ$.

   Therefore, in polar form, it is simply: $10 \angle 60^\circ$

2. Rectangular Form:
   To convert this to rectangular form, use the conversion formulas: $x = r \cos \theta$ $y = r \sin \theta$ where $r = 10$ and $\theta = 60^\circ$.

   • $x = 10 \cos 60^\circ = 10 \times 0.5 = 5$
   • $y = 10 \sin 60^\circ = 10 \times \frac{\sqrt{3}}{2} = 5\sqrt{3} \approx 8.66$

   Thus, in rectangular form, the complex number is: $5 + 8.66j$

Summary:

• Polar Form: $10 \angle 60^\circ$
• Rectangular Form: $5 + 8.66j$

Would you like more details or have any questions?

Here are 5 related questions that could deepen your understanding:

1. How would the rectangular form change if the angle were different, like $45^\circ$?
2. What would be the rectangular form of $10 \cdot e^{j120^\circ}$?
3. How do you convert a complex number from rectangular to polar form?
4. What is the significance of the angle in polar coordinates for complex numbers?
5. How would you interpret the magnitude of a complex number in polar form?

Tip: When converting between forms, always ensure the angle is in the correct unit (degrees or radians) based on your calculations.