Calculate the resistance R, impedance Z, and phase angle from the given circuit diagram and information.

$$X = \frac{3}{2} \times 30$$ $$X = \boxed{\text{45 } \Omega}$$

c) (i)
The sketch shows two sinusoidal waveforms.

• The waveform labeled $I_0 \sin(\omega t)$ starts at zero and increases, representing a sine function.
• The waveform labeled $I_0 \cos(\omega t)$ starts at its maximum positive value and decreases, representing a cosine function. These two waveforms are out of phase by 90 degrees ($\frac{\pi}{2}$ radians), with the sine wave lagging the cosine wave by $90^\circ$ or the cosine wave leading the sine wave by $90^\circ$.

c) (ii)
I.

$$\cos \phi = \frac{V_R}{V_T} = \frac{120}{220}$$ $$\cos \phi \approx 0.545$$

$$\phi = \cos^{-1}(0.545)$$ $$\phi \approx \boxed{\text{57}^\circ}$$

II.

Assuming $R = 20 \, \Omega$ (from a previous part of the question not shown).

$$R = Z \cos \phi$$

$$Z = \frac{4400}{120} = \frac{110}{3}$$ $$Z \approx \boxed{\text{36.67 } \Omega}$$