Question

Find in radians, degrees and grades the angle between the hour-hand and the minute-hand of a clock at
1. half-past three,
2. twenty minutes to six,
3. a quarter past eleven.

Answer 1

Hint: We need to calculate the angle travelled by the hour hand and the minute hand. The difference in these two angles is the angle between the hour-hand and the minute-hand of a clock.
Half past three means \[3.30\;am/pm\]
Twenty minutes to six means \[5.40\;am/pm\]
A quarter past eleven means \[11.15{\text{ }}am/pm\]

Complete step-by-step answer:
We need to find in radians, degrees and grades the angle between the hour-hand and the minute-hand of a clock at half-past three, twenty minutes to six, a quarter past eleven.
Since the hour hand covers a full round in \[12\] hour.
So the hour hand covers \[360^\circ \] in \[12\] hour.
Thus we can say, the hour hand covers \[\dfrac{{360}}{{12}} = 30^\circ \]in \[1\] hour.
We know, \[1\]hour=\[60\] minutes.
Thus the hour hand covers \[30^\circ \]in \[60\] minutes.
The hour hand covers \[\dfrac{{30}}{{60}} = {\left( {\dfrac{1}{2}} \right)^\circ }\]in \[1\] minutes.
(1)The hour hand travelled from \[12\] O’clock to half past three in degree =
\[30^\circ \times 3 + {\left( {\dfrac{1}{2}} \right)^\circ } \times 30 = 90 + 15 = 105^\circ \]
Since the minute hand covers a full round in \[60\] minutes.
So the minutes hand covers \[360^\circ \] in \[60\] minutes.
The minutes hand covers \[\dfrac{{360^\circ }}{{60}} = 6^\circ \]in \[1\] minutes.
The minute hand travelled from \[12\] O’clock to \[30\] minutes making the total angle in degree =\[6^\circ \times 30 = 180^\circ \]
Hence, the angle between the hour-hand and the minute-hand of a clock at
\[(1)\] half-past three is
\[180^\circ - 105^\circ = 75^\circ \]
We know,\[1\] Degree\[ = \dfrac{\pi }{{180}}\]radian
Hence, the angle between the hour-hand and the minute-hand of a clock at \[(1)\] half-past three in radian is \[\dfrac{\pi }{{180}} \times 75 = \dfrac{{5\pi }}{{12}}\]Radian.
Again we know, \[\dfrac{{{\text{Degree}}}}{{{\text{90}}}}{\text{ = }}\dfrac{{{\text{Grade}}}}{{{\text{100}}}}\]
Hence, the angle between the hour-hand and the minute-hand of a clock at \[(1)\] half-past three in grade is \[\dfrac{{75}}{{90}} \times 100 = \dfrac{{250}}{3}\]grade.
(2) The hour hand travelled from 12 O’clock to twenty minutes to six in degree =
\[30^\circ \times 5 + {\left( {\dfrac{1}{2}} \right)^\circ } \times 40 = 150 + 20 = 170^\circ \]
Since the minute hand covers a full round in \[60\] minutes.
So the minutes hand covers \[360^\circ \] in \[60\] minutes.
The minutes hand covers \[\dfrac{{360^\circ }}{{60}} = 6^\circ \]in \[1\] minutes.
The minute hand travelled from \[12\] O’clock to \[40\] minutes making the total angle in degree =\[6^\circ \times 40 = 240^\circ \]
Hence, the angle between the hour-hand and the minute-hand of a clock at twenty minutes to six is
\[240^\circ - 170^\circ = 70^\circ \]
We know,\[1\]Degree\[ = \dfrac{\pi }{{180}}\]radian
Hence, the angle between the hour-hand and the minute-hand of a clock at twenty minutes to six in radian is \[\dfrac{\pi }{{180}} \times 70 = \dfrac{{7\pi }}{{18}}\] Radian.
Again we know, \[\dfrac{{Degree}}{{90}} = \dfrac{{Grade}}{{100}}\]
Hence, the angle between the hour-hand and the minute-hand of a clock at twenty minutes to six in grade is \[\dfrac{{70}}{{90}} \times 100 = \dfrac{{700}}{9}\] grade.
(3)The hour hand travelled from 12 O’clock to a quarter past eleven in degree =
\[30^\circ \times 11 + {\left( {\dfrac{1}{2}} \right)^\circ } \times 15 = 330 + \dfrac{{15}}{2} = \dfrac{{660 + 15}}{2} = {\left( {\dfrac{{675}}{2}} \right)^\circ }\]
Since the minute hand covers a full round in \[60\] minutes.
So the minutes hand covers \[360^\circ \]in \[60\] minutes.
The minutes hand covers \[\dfrac{{360^\circ }}{{60}} = 6^\circ \] in \[1\] minutes.
The minute hand travelled from \[12\] O’clock to \[15\] minutes making the total angle in degree =\[6^\circ \times 15 = 90^\circ \]
Hence, the angle between the hour-hand and the minute-hand of a clock at a quarter past eleven is
\[{\left( {\dfrac{{675}}{2}} \right)^\circ } - 90^\circ = \dfrac{{675 - 180}}{2} = {\left( {\dfrac{{495}}{2}} \right)^\circ }\]
We know,\[1\] Degree \[ = \dfrac{\pi }{{180}}\] radian
Hence, the angle between the hour-hand and the minute-hand of a clock at a quarter past eleven in radian is \[\dfrac{\pi }{{180}} \times \dfrac{{495}}{2} = \dfrac{{11\pi }}{8}\] Radian.
Again we know, \[\dfrac{{{\text{Degree}}}}{{{\text{90}}}}{\text{ = }}\dfrac{{{\text{Grade}}}}{{{\text{100}}}}\]
Hence, the angle between the hour-hand and the minute-hand of a clock at a quarter past eleven in grade is \[\dfrac{{\dfrac{{495}}{2}}}{{90}} \times 100 = \dfrac{{495}}{{90 \times 2}} \times 100 = 55 \times 5 = 275\]grade.

Note: An hour is most commonly defined as a period of time equal to \[60\] minutes, where a minute is equal to \[60\] seconds, and a second has a rigorous scientific definition. There are also \[24\] hours in a day. Most people read time using either a \[12\]-hour clock or a \[24\]-hour clock.
\[12\]-hour clock: A \[12\]-hour clock uses the numbers \[1 - 12\].
\[24\]-hour clock: A \[24\]-hour clock uses the numbers \[0 - 23\].
Relation between degree, radian and grade is
\[\dfrac{{{\text{Degree}}}}{{{\text{90}}}}{\text{ = }}\dfrac{{{\text{Grade}}}}{{{\text{100}}}}{\text{ = }}\dfrac{{{\text{2Radian}}}}{{{ \pi }}}\]