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

Answer
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Hint: Find the time from the given statement. First find the angle made by the hour hand from 120clock 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 π180.
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 30clock. 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 360or 2π.
A clock has a total of 12 hours shown in it. Thus we can write that, 360=12hrs.
Now let us find the angle made by the clock in 1 hour.
1hr=36012=30 1hr=30 
Similarly, we know that 1hr=60min. Hence we can change the above expression as,
60min=30.
Thus for 60 min it is 30, so for 1 minute it will be,
1min=6030=12
Now we found the time as 3:30. Here the hour hand travels 3 hours and 30 minutes from 120clockat half past three.
Thus, the hour hand travels= 3 hours and 30 minutes
3hrs+30min=(3×1hr)+(30×1min)
We found out that, 1hr=30and 1min=12, thus substitute these values and simplify it.
(3×1hr)+(30×1min)=(3×30)+(30×12) (3×30)+(30×12)=90+15=105 
Thus the total angle made by the hour hand=105(1).
Similarly let us find the angle made by the minute’s hand.
Now again we know that, 360=60min
Thus for 1min the angle is,
1min=36060=6 1min=6  
Hence in 30 minutes the angle is,
30min=30×1min 30min=30×6 30min=180(2) 
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)= 180105=75
Thus we got the angle between the hour hand and minutes hands as 75, which is in degrees.
Now let us convert the given angle of degrees to radians. To convert degrees to radians, multiply the angle with π180. Let us consider the angle as equal to θ. Hence, θ=75. Now let us convert it to radians.
θ=75=(75×π180) θ=75×π180=5π12  
Thus we the angle between the hour hand and minute hand as,
In degrees, θ=75.
In radians, θ=5π12.
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 360or 2π, which is the same as in a circle.

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