The Polarized Windshield Mystery · AI Generator #GEN-478F3AE7
Physics · optics
Problem
A driver notices that when wearing polarized sunglasses, the digital display on their car's dashboard appears completely dark when viewed at a certain angle, but becomes visible again when they tilt their head. Investigation reveals that the LCD display emits light polarized at 30° from the vertical, while the sunglasses have a polarizing filter oriented vertically.
The driver's eye is initially positioned such that the line of sight makes an angle θ with the normal to the dashboard display. When the driver tilts their head by an angle ϕ=60° (rotating about the line of sight), the transmitted light intensity becomes exactly half of what it would be without any polarizing filter.
Find the initial viewing angle θ at which the display appeared completely dark.