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I'm studying the clock angle problem. It's a simple problem, you can read the description, the analysis and the solution in 3 to 5 minutes here:

Clock angle problem - Wikipedia

There is a detail that I don't understand, in the hour angle calculation. The formula is:

hour angle = 0.5° * (60 * H + M)

where H = hour and M = minutes

so, if we take for example 3:27:

hour angle = 0.5 * (50 * 3 + 27) = 103.5

I don't understand why we add the minutes ? Why does the angle of the hour hand depends of the minutes ? In my mind, it depends only of the distance between 12 o'clock and the hour hand.

Thank you very much for your insights

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The hour hand moves each minute; it doesn't jump straight between the numbers at every hour. For every minute, it moves $\frac{1}{60}$ of the distance between two of the clock numbers. There are $\frac{360^\circ}{12} = 30^\circ$ in each of these sections; thus, each minute, the hour hand moves $0.5$ degrees.

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    $\begingroup$ It's so obvious now than you said it ^^ $\endgroup$ – Vivien Adnot Oct 15 '17 at 9:47
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Hour hand moves continuously with every minute, every second, every nanosecond, and so on... even if it's not noticeable when you look at a clock (not digital of course). Unless you have a clock whose hour hand moves from $3$ to $4$ at $3:59:59$, it has to depend on minutes , seconds, etc.

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