An incident ray of light is initially normal to the surface of a plane mirror. The mirror is rotated until the angle between the incident and reflected rays is 30°. Through what angle has the mirror been rotated?

Hello!

If an incident ray was normal to the plane mirror, then its reflected ray was normal also. In this case, the mirror has been rotated, and the incident ray remains the same, as I understand. The angle between the incident ray and the new reflected ray has become 30 degrees. Denote it as `alpha.` Denote the angle of mirror rotation as `beta.`

Please look at the picture. The red line (the initial mirror position) is perpendicular to the green line (an incident ray). The orange line (the new mirror position) is perpendicular to the dashed green line (the new normal). Also, the new normal bisects the angle between the green and blue lines (the incident ray and the new reflected ray).

Now we can find `beta.` We know that the angle between the blue line and the red line is `90-beta-alpha/2` and it is also `90-alpha.` From the equation `90-beta-alpha/2 = 90-alpha` we obtain `beta=alpha/2=15` degrees. This is the answer.

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