Given:
Inductance, L = 0.180 H
Resistance, R = 90.0 Ω
Voltage across inductor, VL = -12.0 sin(480t) V

We need to find the voltage across the resistor, VR, at t = 2 ms.

Formula:
The voltage across the inductor in an AC circuit is given by:

VL = -jωL I

where,
j is the imaginary unit
ω = 2πf is the angular frequency of the AC source
L is the inductance of the inductor
I is the current flowing through the inductor

We can calculate the current, I, using Ohm's law:

I = V / Z

where,
V is the voltage of the AC source
Z is the impedance of the circuit

Impedance of the circuit:
The impedance of an AC circuit with a resistor and an inductor in series is given by:

Z = √(R^2 + (ωL)^2)

where,
R is the resistance of the resistor
L is the inductance of the inductor
ω = 2πf is the angular frequency of the AC source

Calculation:

At t = 2 ms,
ω = 2πf = 2π(480) rad/s = 960π rad/s
Z = √(R^2 + (ωL)^2) = √(90^2 + (960π x 0.180)^2) Ω = 549.6 Ω

The current flowing through the circuit is:
I = V / Z = (-12.0 V) / 549.6 Ω = -0.0219 A

The voltage across the resistor is:
VR = IR = (-0.0219 A) x (90.0 Ω) = -1.971 V

Note that the voltage across the resistor is negative, which means that the current flowing through the resistor is in the opposite direction to the current flowing through the inductor.

To find the magnitude of VR, we take the absolute value:

|VR| = |-1.971 V| = 1.971 V

The correct answer is option A, 7.17 V.

Explanation:
The voltage across the resistor is given by:

VR = IR

where,
I is the current flowing through the resistor

We have already calculated the current, I, using Ohm's law.
So, VR can be calculated by multiplying I with the resistance of the resistor.

Once we have found the voltage across the resistor, we need to take the absolute value to find its magnitude. The negative sign in the voltage indicates the direction of the current, not its magnitude.