Trigonometry question please help

Question

The positive measure of the angle between the hour hand and the minute hand at half past two equals ...

Please help me I don’t understand 🥹

Thank you so much in advance 🥹

Answer 1

105 degrees

= the angle between minute and hour hands at 2:30

hour hand is between 2 and 3

minute hand is at 6

from 3 to 6 is 1/4 of the circumference

from 3 to half way to 2 is 1/24 of the circumference

1/4 + 1/24 = 7/24 of 2pi radians

= 14pi/24 = 7pi/12 radians

= 7pi/12 x 180/pi = 7/12x180 = 105 degrees

Answer 2

Half past 2 is 2:30:

The hour hand is halfway between 2 & 3. If it were at 2:00, it would be 2/12 of the way around the clock and if it were at 3:00, it would be at 3/12 of its way around the clock. Halfway would be 2.5/12 or 5/24 of the way around the clock.

The minute hand is halfway around the clock or at 12/24 of its way around (using 24 because its a common denominator with what I got from the hour hand).

So the "angle" between them is 12/24 - 5/24 = 7/24. To convert that to radians, since there are 2π radians in the complete clock cycle: (7/24)(2π) = 7π/12 radians.

To convert to degrees: (7π/12)(180/π) = 7•180/12 = 105°