To convert the voltage with a magnitude of $100 \, \text{V}$ and a phase angle of $45^\circ$ into a complex number, we can express it in the form:

$V = V_{\text{magnitude}} \cdot (\cos(\theta) + i \sin(\theta))$

where:

• $V_{\text{magnitude}} = 100 \, \text{V}$
• $\theta = 45^\circ$

Steps:

1. Calculate the Real Part: $\text{Real part} = 100 \cdot \cos(45^\circ) = 100 \cdot \frac{\sqrt{2}}{2} = 100 \cdot 0.7071 = 70.71$

2. Calculate the Imaginary Part: $\text{Imaginary part} = 100 \cdot \sin(45^\circ) = 100 \cdot \frac{\sqrt{2}}{2} = 100 \cdot 0.7071 = 70.71$

Thus, the voltage in complex form is:

$V = 70.71 + 70.71i \, \text{V}$

or, in polar form:

$V = 100 \angle 45^\circ$

Would you like further details on complex conversions, or do you have any other questions?

Here are five related questions to explore further:

1. How do you convert a complex number in rectangular form back to polar form?
2. What is the significance of phase angles in AC circuits?
3. How would you interpret this voltage in terms of sinusoidal waveforms?
4. What role does the capacitor play in affecting phase angles in AC circuits?
5. How do impedance and phase angle relate in an AC circuit with capacitive elements?

Tip: Remember, for phase angles, converting degrees to radians (or vice versa) may be necessary, especially in calculations involving trigonometric functions in different contexts.