In our last department meeting, one teacher presented a puzzle/problem for us to figure out.
At 3:00, the hour and minute hands on a clock form a right angle. What is the next time that happens?
The presenting teacher had a pretty darn elegant solution. But I enjoyed working it out using brute force. (That’s pretty much my go-to.) I’m going to type my solution down below the jump.
I saw the hour hand and the minute hand act like vectors coming out of the origin. Their position changed in time.
and , where is measure in minutes.
Now when are two vectors perpendicular? When their dot product is 0.
And lo and behold, naturally occurring in this problem, we see a difference of angles formula for cosine!
And then we just have to find all values that make this true. Well the cosine function has an output of 0 when the input of the cosine function is (where is an integer).
That just leaves us with:
From this, we solve for and we get:
Let’s see what happens for various values of !
And so here we have our answer… around 212 minutes (3:32 and 44 seconds).
Pretty neat, huh? Okay, okay, once you look at the final solution for the general form for all the times, you might suspect there is a much easier thing going on here. And that’s what the teacher presenting the problem shared with us after we finished. But I’m not going to share it with you because I LIKE MY SOLUTION and HECK IF I DIDN’T SEE AND USE A SUM OF ANGLES TRIG FORMULA IN THE WILD! Huzzah!