Answer :
Certainly! Let's find the positive radian measure of the larger angle formed by the hands of a clock at 1 o'clock. Here’s a detailed step-by-step solution:
1. Determine the positions of the hour and minute hands:
- At 1 o'clock, the minute hand is pointing at the 12.
- The hour hand at 1 o'clock is pointing at the 1.
2. Calculate the degree measure of each hour on a clock:
- Since a full circle is 360 degrees and there are 12 hours on a clock, each hour represents:
[tex]\[ \frac{360}{12} = 30 \text{ degrees} \][/tex]
- Therefore, the hour hand at the 1 is at:
[tex]\[ 1 \times 30 = 30 \text{ degrees} \][/tex]
3. Determine the minute hand’s position in degrees:
- At the 12, the minute hand is at 0 degrees.
4. Calculate the smaller angle between the hour and minute hands:
- Since the hour hand is at 30 degrees and the minute hand is at 0 degrees, the smaller angle is:
[tex]\[ |30 - 0| = 30 \text{ degrees} \][/tex]
5. Calculate the larger angle between the hour and minute hands:
- The larger angle is the complement to 360 degrees:
[tex]\[ 360 - 30 = 330 \text{ degrees} \][/tex]
6. Convert the larger angle from degrees to radians:
- To convert degrees to radians, use the fact that 180 degrees equals [tex]\(\pi\)[/tex] radians:
[tex]\[ 330 \text{ degrees} \times \frac{\pi \text{ radians}}{180 \text{ degrees}} = \frac{330\pi}{180} \text{ radians} \][/tex]
- Simplify the fraction [tex]\(\frac{330\pi}{180}\)[/tex]:
[tex]\[ \frac{330\pi}{180} = \frac{11\pi}{6} \text{ radians} \][/tex]
Therefore, the positive radian measure of the larger angle formed by the hands of a clock at 1 o'clock is:
[tex]\[ \boxed{\frac{11\pi}{6} \ \text{radians}} \][/tex]
1. Determine the positions of the hour and minute hands:
- At 1 o'clock, the minute hand is pointing at the 12.
- The hour hand at 1 o'clock is pointing at the 1.
2. Calculate the degree measure of each hour on a clock:
- Since a full circle is 360 degrees and there are 12 hours on a clock, each hour represents:
[tex]\[ \frac{360}{12} = 30 \text{ degrees} \][/tex]
- Therefore, the hour hand at the 1 is at:
[tex]\[ 1 \times 30 = 30 \text{ degrees} \][/tex]
3. Determine the minute hand’s position in degrees:
- At the 12, the minute hand is at 0 degrees.
4. Calculate the smaller angle between the hour and minute hands:
- Since the hour hand is at 30 degrees and the minute hand is at 0 degrees, the smaller angle is:
[tex]\[ |30 - 0| = 30 \text{ degrees} \][/tex]
5. Calculate the larger angle between the hour and minute hands:
- The larger angle is the complement to 360 degrees:
[tex]\[ 360 - 30 = 330 \text{ degrees} \][/tex]
6. Convert the larger angle from degrees to radians:
- To convert degrees to radians, use the fact that 180 degrees equals [tex]\(\pi\)[/tex] radians:
[tex]\[ 330 \text{ degrees} \times \frac{\pi \text{ radians}}{180 \text{ degrees}} = \frac{330\pi}{180} \text{ radians} \][/tex]
- Simplify the fraction [tex]\(\frac{330\pi}{180}\)[/tex]:
[tex]\[ \frac{330\pi}{180} = \frac{11\pi}{6} \text{ radians} \][/tex]
Therefore, the positive radian measure of the larger angle formed by the hands of a clock at 1 o'clock is:
[tex]\[ \boxed{\frac{11\pi}{6} \ \text{radians}} \][/tex]
