Wikipedia:Reference desk/Archives/Mathematics/2021 May 28

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May 28

Mathematics

Variation's and the types. — Preceding unsigned comment added by 105.112.116.231 (talk) 07:36, 28 May 2021 (UTC)

Is there a question hidden in here?  --Lambiam 11:15, 28 May 2021 (UTC)

Is there a formula for the angle between the hour and minute hands on a clock?

I think it would involve sine. I don't see the formula that I believe I am looking for in Clock angle problem.

12 Midnight would be zero, 6 A.M. would be π and 12 Noon would be 2π.— Vchimpanzee • talk • contributions • 22:41, 28 May 2021 (UTC)

There is no trigonometry involved in this problem. At h hours past midnight, the hour hand has rotated ${\frac {h}{12}}\cdot 2\pi$ clockwise from vertical and the minute hand has rotated $h\cdot 2\pi$ clockwise from vertical, so the angle between them (measured clockwise from the hour hand) is ${\frac {11h}{12}}\cdot 2\pi {\pmod {2\pi }}$. --JBL (talk) 22:49, 28 May 2021 (UTC)

Maybe not the sine function as such, but I'm picturing a curve similar to the sine.— Vchimpanzee • talk • contributions • 16:05, 29 May 2021 (UTC)

The graph of the angle does not curve at all; depending what exactly you want to measure, it is either a sawtooth wave (if you measure the angle in a fixed orientation, so that it always lies in the interval $[0,2\pi )$ -- that's what the formula above gives) or a triangle wave (if you don't care about orientation, so that the angle always lies in the interval $[0,\pi ]$). --JBL (talk) 16:13, 29 May 2021 (UTC)

I forgot one very important detail. The hands will cross so the minute hand is below or to the right of the hour hand. I was interpreting that as a negative angle until it reached -π. Actually, that wouldn't work.

Anyway, someone (I guess the props department is to blame) put a clock in a scene in a TV series where the minute and hour hands appeared to be the same length. It was either 5:00 or 12:25, but of course if it was 12:25 the hour hand would have moved off the 12. And the minute hand moved once a minute and we saw it do that several times.— Vchimpanzee • talk • contributions • 16:22, 29 May 2021 (UTC)

(edit conflict) It is not clear to me whether or not you understand my first comment. If you do, you can adapt the same reasoning to any particular method you like of measuring the angles. If not, I would be happy to elaborate; it would help if you made a clear indication of what parts you do and do not understand. It is not clear to me how your latest comment relates to your original question. --JBL (talk) 16:27, 29 May 2021 (UTC)

I guess there's not really a way to explain the connection. On two separate TV episodes I thought the angle looked weird, but there's no way short of having the clocks in question right in front of me and measuring the angles with a protractor and then using your formula.— Vchimpanzee • talk • contributions • 16:56, 29 May 2021 (UTC)

At this point I have trouble with visualizing the angles. I can try the numbers but unless I can measure the actual angles, that's a problem. If anyone wants to try this, both TV episodes are available online. Maybe there's some computer software which, if the video is paused, could actually measure the angles. If all that's too complicated, I'll just try to work with what I have. I think in a lot of cases I'm just dividing 60 by 11 and working from there.— Vchimpanzee • talk • contributions • 17:20, 29 May 2021 (UTC)

All angles, from $0^{\circ }$ to $360^{\circ }$, occur once about every hour and five minutes. If you have measured the angle in an image you found somewhere, take the graph in the section Clock angle problem § When are the hour and minute hands of a clock superimposed?, find the point representing that angle on the x-axis at T = 12, and draw a diagonal line through that point, sloping downwards at a $45^{\circ }$ angle, and wrapping around at the vertical margins at $0^{\circ }$ and $360^{\circ }$, like the red lines in the graph. Each point where this diagonal line crosses a blue line gives a combination of the hour hand and minute hand that produce that angle. There are always 11 crossing points. The only thing that can be weird is the combination of the two angles made by the hands with a fixed reference line, say the vertical.  --Lambiam 22:08, 29 May 2021 (UTC)

There are some times when the same hands could indicate two different times (depending on which one is the minute and which one is the hour hand): [1]. (5:00 is not one of them, but there is a time just before 5 that is indistinguishable from roughly 11:24, and there is a time just after 5 that is indistinguishable from roughly 12:26.)