I encountered this formula, $x=\frac{11m}2-30h$, where x is the angle between the minute hand and hour hand of a clock. m being minutes and h being hours. Ex- at 3:20, m=20 and h=3, so, x=20 degree from the above formula. It's a very handy formula. But I wonder how to derive it.
I see that minute hand makes 6 degrees in 1 minute and hour hand makes half degree. So, the difference in their angles after m minutes will be $\frac{11m}2$. But from where did we get 30h?
Best Answer
As the hour hand, in $1$ minutes rotates $\displaystyle\frac12^\circ,$
So, in $h$ hour $m$ minute $\displaystyle =60h+m$ minute it will rotate $\displaystyle \left(\frac{60h+m}2\right)^\circ=\left(30h+\frac m2\right)^\circ$
Similarly, in $h$ hour $m$ minute the minute hand will rotate $\displaystyle 6(60h+m)^\circ\equiv 6m^\circ\pmod{360^\circ}$ assuming $h$ to be some integer
Now find the difference