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

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
$\dfrac{{{\text{Degree}}}}{{{\text{90}}}}{\text{ = }}\dfrac{{{\text{Grade}}}}{{{\text{100}}}}{\text{ = }}\dfrac{{{\text{2Radian}}}}{{{ \pi }}}$