What angle (in degrees) do the minute and hour hands make when the clock shows exactly 8 o'clock?
The solution of the problem
This lesson shows how to use the properties of a circle in problems with a clock face (determining the angles between the hour and minute hands). When solving the problem, we use the property of a circle: a complete revolution of a circle is 360 degrees. Considering that the dial is divided into 12 equal hours, you can easily determine how many degrees correspond to one hour. The further solution comes down to correctly determining the difference in hours between the minute and hour hands, and performing simple multiplication. When solving problems, it should be clearly understood that we are considering the position of the hour and minute hands relative to their position to the clock cutoffs, i.e. from 1 to 12.
The solution to this problem is recommended for 7th grade students when studying the topic “Triangles” (“Circle. Typical problems”), for 8th grade students when studying the topic “Circle” (“Relative position of a straight line and a circle”, “Central angle. Degree measure of the arc of a circle"), for 9th grade students when studying the topic “Circle length and area of a circle” (“A circle circumscribed about a regular polygon”). When preparing for the OGE, the lesson is recommended for reviewing the topics “Circumference”, “Circle Length and Area of a Circle”.
Hour angle
dihedral angle between the planes of the celestial meridian and the circle of declination, one of the equatorial coordinates in astronomy. Usually counted in hourly units in both directions from the southern part of the celestial meridian (from 0 to +12 hours to the west and to -12 hours to the east).
Astronomical Dictionary. EdwART. 2010.
See what “Hour angle” is in other dictionaries:
Big Encyclopedic Dictionary
The celestial coordinate system is used in astronomy to describe the position of luminaries in the sky or points on an imaginary celestial sphere. The coordinates of luminaries or points are specified by two angular values (or arcs), which uniquely determine the position... ... Wikipedia
The dihedral angle between the planes of the celestial meridian and the circle of declination, one of the equatorial coordinates in astronomy. Usually counted in hourly units in both directions from the southern part of the celestial meridian (from 0 to +12 hours to the west and up to 12 hours to ... ... encyclopedic Dictionary
hour angle- valandų kampas statusas T sritis fizika atitikmenys: engl. hour angle vok. Stundenwinkel, m rus. hour angle, m pranc. angle horaire, m … Fizikos terminų žodynas
The dihedral angle between the planes of the celestial meridian and the circle of declination, one of the equatorial coordinates in astronomy. Usually measured hourly in both directions from the south. parts of the celestial meridian (from 0 to + 12 hours to 3. and up to 12 hours to E.) ... Natural science. encyclopedic Dictionary
One of the coordinates in the equatorial celestial coordinate system; standard designation t. See Celestial Coordinates... Great Soviet Encyclopedia
See Celestial Coordinates... Big Encyclopedic Polytechnic Dictionary
Let us turn again to school tasks and intelligence tasks. One of these tasks is to find out what angle the minute and hour hands form between themselves on a mechanical watch at 16 hours 38 minutes, or one of the variations is to find out how much time it will be after the beginning of the first day when the hour and minute hands form an angle of 70 degrees.
Or in the most general terms "find the angle between the hour and minute hands"(With)
The simplest question to which many people manage to give the wrong answer. What is the angle between the hour and minute hands on a clock at 15:15?
The answer zero degrees is not the correct answer :)
Let's figure it out.
In 60 minutes, the minute hand makes a full revolution around the dial, that is, it rotates 360 degrees. During the same time (60 minutes), the hour hand will travel only one-twelfth of the circle, that is, it will move by 360/12 = 30 degrees
As for the minute, everything is very simple. Compiling proportion minutes are related to the angle traversed as a complete revolution (60 minutes) is to 360 degrees.
Thus, the angle traveled by the minute hand will be minutes/60*360 = minutes*6
As a result, the conclusion Each minute passed moves the minute hand 6 degrees
Great! Now what about the sentry. But the principle is the same, only you need to reduce the time (hours and minutes) to fractions of an hour.
For example, 2 hours 30 minutes is 2.5 hours (2 hours and half), 8 hours and 15 minutes is 8.25 (8 hours and one quarter of an hour), 11 hours 45 minutes is 11 hours and three quarters of an hour, that is, 8.75)
Thus, the angle traversed by the clock hand will be hours (in fractions of an hour) * 360.12 = hours * 30
And as a consequence the conclusion Every hour passed moves the hour hand 30 degrees
angle between hands = (hour+(minutes /60))*30 -minutes*6
Where hour+(minutes /60)- this is the clockwise position
Thus, the answer to the problem: what angle will the hands make when the clock shows 15 hours 15 minutes, will be as follows:
15 hours 15 minutes is equivalent to the position of the hands at 3 hours and 15 minutes and thus the angle will be (3+15/60)*30-15*6=7.5 degrees
Determine the time by the angle between the arrows
This task is more difficult, since we will solve it in a general form, that is, determine all pairs (hour and minute) when they form a given angle.
So, let's remember. If time is expressed as HH:MM (hour:minute) then the angle between the hands is expressed by the formula
Now, if we denote the angle by the letter U and convert everything into an alternative form, we get the following formula
Or, getting rid of the denominator, we get the basic formula relating the angle between two hands and the positions of these hands on the dial.
note that the angle can also be negative, i.e. oh, within an hour we can meet the same angle twice, for example, an angle of 7.5 degrees can be at 15 hours 15 minutes and 15 hours and 17.72727272 minutes
If, as in the first problem, we were given an angle, then we get an equation with two variables. In principle, it cannot be solved unless one accepts the condition that the hour and minute can only be integers.
Under this condition we obtain the classical Diophantine equation. The solution to which is very simple. We will not consider them for now, but will immediately present the final formulas
where k is an arbitrary integer.
We naturally take the result of hours modulo 24, and the result of minutes modulo 60
Let's count all the options when the hour and minute hands coincide? That is, when the angle between them is 0 degrees.
At a minimum, we know two such points: 0 hours and 0 minutes and 12 noon 0 minutes. What about the rest??
Let's create a table showing the positions of the arrows when the angle between them is zero degrees
Oops! on the third line we have an error at 10 o'clock, the hands do not match. This can be seen by looking at the dial. What's the matter?? It seems like everything was calculated correctly.
But the whole point is that in the interval between 10 and 11 o’clock, in order for the minute and hour hands to coincide, the minute hand must be somewhere in the fractional part of a minute.
This can be easily checked using the formula by substituting the number zero instead of the angle, and the number 10 instead of the hour.
we get that the minute hand will be located between (!!) divisions 54 and 55 (exactly at the position 54.545454 minutes).
That's why our latest formulas didn't work, since we assumed that hours and minutes are integers(!).
Problems that appear on the Unified State Exam
We will look at problems for which solutions are available on the Internet, but we will take a different route. Perhaps this will make it easier for that part of schoolchildren who are looking for a simple and easy way to solve problems.
After all, the more different options for solving problems, the better.
So, we know only one formula and we will only use it.
The clock with hands shows 1 hour 35 minutes. In how many minutes will the minute hand line up with the hour hand for the tenth time?
The reasoning of the “solvers” on other Internet resources made me a little tired and confused. For those “tired” like me, we solve this problem differently.
Let’s determine when in the first (1) hour the minute and hour hands coincide (angle 0 degrees)? We substitute the known numbers into the equation and get
that is, 1 hour and almost 5.5 minutes. is it earlier than 1 hour 35 minutes? Yes! Great, then we don’t take this hour into account in further calculations.
We need to find the 10th coincidence of the minute and hour hands, we begin to analyze:
for the first time the hour hand will be at 2 o'clock and how many minutes,
the second time at 3 o'clock and how many minutes
for the eighth time at 9 o'clock and for some minutes
for the ninth time at 10 o'clock and how many minutes
for the ninth time at 11 o'clock and for some minutes
Now all that remains is to find where the minute hand will be at 11 o'clock, so that the hands coincide
And now we multiply 10 times the revolution (which is every hour) by 60 (converting to minutes) and we get 600 minutes. and calculate the difference between 60 minutes and 35 minutes (which were specified)
The final answer was 625 minutes.
Q.E.D. There is no need for any equations, proportions, or which of the arrows moved at what speed. It's all tinsel. It is enough to know one formula.
A more interesting and complex task sounds like this. At 8 pm, the angle between the hour and minute hands is 31 degrees. How long will the hand show the time after the minute and hour hands form a right angle 5 times?
So in our formula, two of the three parameters are again known: 8 and 31 degrees. We determine the minute hand using the formula and get 38 minutes.
When is the nearest time when the arrows will form a right (90 degrees) angle?
That is, at 8 hours 27.27272727 minutes this is the first right angle in this hour and at 8 hours and 60 minutes this is the second right angle in this hour.
The first right angle has already passed relative to the given time, so we do not count it.
The first 90 degrees at 8 hours 60 minutes (we can say that exactly at 9-00) - once
at 9 o'clock and how many minutes - that's two
at 10 o'clock and how many minutes is it three
again at 10 and how many minutes is 4, so there are two coincidences at 10 o’clock
and at 11 o'clock and how many minutes is five.
It’s even easier if we use a bot. Enter 90 degrees and get the following table
| Time on the dial when the specified angle will be | ||||||||||||||||||||||||||||||||||||||||||||||||||
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that is, at 11 hours 10.90 minutes there will be just the fifth time when a right angle is again formed between the hour and minute hands.