Calculating the Smallest Angle of an Analogue Clock

I don’t know what I’m doing with my life

What is the smallest angle that can be formed by the

hour hand and the minute hand at 15:40?

At 15:40 the

minute hand points to 8

At 15:40 the hour hand points to the inbetween 3 and 4

At 8, the minute hands is on 2/3 rotation, so the angle is 2/3 of 360°

And that is 240°

40 minutes is 2/3 of an

hour, there’s a 30° rotation in an hour, so there’s 2/3 of 30° rotation in 40 minutes and that is 20° in the

inbetween of 3 and 4.

adding the angle of the hour hand at 3 and that 20° from earlier,

we got 90° + 20° = 110°

By subtracting the minute hand’s angle by the hour hand’s angle, we got the smallest angle that could be formed at 15:40

We got 240°-110°=130°

� =sin

^{− 1}

( ^{sin ( 15 2 3 ∙ 3 600 2 � )} )

for the second hand

 

for the minute hand

 

for the hour hand

 

15:40 is hour, in second there’s seconds

 

 

� =sin

⁻¹

( ^{sin ( 15 2 3 ∙ 12 2 ∙ 3 600 � )} )