i) You are testing the voltage across a capacitor in an AC circuit. The instrument you are using indicates this voltage to have a magnitude of 100 V and a phase angle of 45 degrees. Convert this voltage into a complex number.
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To convert the given voltage into a complex number, we start with the polar form: \( V = 100 \angle 45^\circ \). We can convert this to rectangular form using the formula \( V = r (\cos \theta + j \sin \theta) \), where \( r \) is the magnitude and \( \theta \) is the phase angle. Calculating the real and imaginary parts: Real part: \( 100 \cos(45^\circ) = 100 \times \frac{\sqrt{2}}{2} \approx 70.71 \) Imaginary part: \( 100 \sin(45^\circ) = 100 \times \frac{\sqrt{2}}{2} \approx 70.71 \) So, the voltage in complex form is \( V \approx 70.71 + j70.71 \) V. Now you have a vibrant complex number representing your voltage, perfect for analyzing in the world of AC circuits!