Question

Find in degrees and radians the angle between the hour hand and minute hand of a clock at half past three.

Answer 1

Hint: Find the time from the given statement. First find the angle made by the hour hand from \[12'0clock\] at half past three. Then find the angle made by the minute’s hand, now subtract these angles to get the angle made by both hour and minute hand in degree. Convert it to radians by multiplying the angle with \[\dfrac{\pi }{{180}}\].
Complete step-by-step answer:
We have to find the angle between the hour hand and minute hand of a clock. We have been given the time as half past three, which means that it is 30 minutes past the hour i.e., it is 30 minutes past \[3'0clock\]. Hence the time is \[3:30\] which is half past three.
We know that a clock is in a circle. Total angle of the circle is \[360^\circ \]or \[2\pi \].
A clock has a total of 12 hours shown in it. Thus we can write that, \[360^\circ = 12hrs\].
Now let us find the angle made by the clock in 1 hour.
\[\begin{gathered}
  1hr = \dfrac{{360^\circ }}{{12}} = 30^\circ \ \\
  1hr = 30^\circ \ \\
\end{gathered} \]
Similarly, we know that \[1hr = 60\min \]. Hence we can change the above expression as,
\[60\min = 30^\circ \].
Thus for 60 min it is \[30^\circ \], so for 1 minute it will be,
\[1\min = \dfrac{{60}}{{30}} = \dfrac{1}{2}\]
Now we found the time as \[3:30\]. Here the hour hand travels 3 hours and 30 minutes from \[12'0clock\]at half past three.
Thus, the hour hand travels= 3 hours and 30 minutes
\[3hrs + 30\min = \left( {3 \times 1hr} \right) + \left( {30 \times 1\min } \right)\]
We found out that, \[1hr = 30^\circ \]and \[1\min = \dfrac{1}{2}\], thus substitute these values and simplify it.
\[\begin{gathered}
  \left( {3 \times 1hr} \right) + \left( {30 \times 1\min } \right) = \left( {3 \times 30} \right) + \left( {30 \times \dfrac{1}{2}} \right) \ \\
  \left( {3 \times 30} \right) + \left( {30 \times \dfrac{1}{2}} \right) = 90 + 15 = 105^\circ \ \\
\end{gathered} \]
Thus the total angle made by the hour hand=\[105^\circ - \left( 1 \right)\].
Similarly let us find the angle made by the minute’s hand.
Now again we know that, \[360^\circ = 60\min \]
Thus for \[1\min \] the angle is,
\[\begin{gathered}
  1\min = \dfrac{{360}}{{60}} = 6^\circ \ \\
  1\min = 6^\circ \ \\
   \ \\
\end{gathered} \]
Hence in 30 minutes the angle is,
\[\begin{gathered}
  30\min = 30 \times 1\min \ \\
  30\min = 30 \times 6^\circ \ \\
  30\min = 180^\circ - \left( 2 \right) \ \\
\end{gathered} \]
Thus the angle between the hour hand the minute hand at half past three= (angle made by the minutes hand) – (angle made by the hour hand)= \[180^\circ - 105^\circ = 75^\circ \]
Thus we got the angle between the hour hand and minutes hands as \[75^\circ \], which is in degrees.
Now let us convert the given angle of degrees to radians. To convert degrees to radians, multiply the angle with \[\dfrac{\pi }{{180}}\]. Let us consider the angle as equal to \[\theta \]. Hence, \[\theta = 75^\circ \]. Now let us convert it to radians.
\[\begin{gathered}
  \theta = 75^\circ = \left( {75 \times \dfrac{\pi }{{180}}} \right) \ \\
  \theta = 75 \times \dfrac{\pi }{{180}} = \dfrac{{5\pi }}{{12}} \ \\
   \ \\
\end{gathered} \]
Thus we the angle between the hour hand and minute hand as,
In degrees, \[\theta = 75^\circ \].
In radians, \[\theta = \dfrac{{5\pi }}{{12}}\].
Thus we got the required values.
Note: The basic of questions like these is that you should know the conversion of the hour and minutes Hand with respect to the angle subtended by the clock. Remember the basics of conversion from hours to minutes. Also keep in mind that the total angle subtended by the clock is \[360^\circ \]or \[2\pi \], which is the same as in a circle.