How often are the clock hands at a right angle?

Question

The hour and minute hands of a clock form a right angle (exactly $90°$ apart) at moments like 3:00 and 9:00, but also at many in-between times.

How many times in a 12-hour period are the two hands exactly $90°$ apart? It's not 24.

Accepted Answer

The hands form a right angle 22 times in 12 hours (44 times in a full day). Track the relative angle. The minute hand gains on the hour hand at 6 - 0.5 = 5.5° per minute. Over 12 hours the minute hand laps the hour hand 12 - 1 = 11 times, meaning the relative angle between them sweeps through a full 360° exactly 11 times. Two right angles per relative lap. During each full 360° of relative motion, the hands are 90° apart at two distinct moments: once when the minute hand is 90° ahead of the hour hand, and once when it's 90° ahead again on the way to lapping (equivalently 270°, which looks lik…

How often are the clock hands at a right angle?

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