Clock angles solution

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This is an example of what I would call a thorough written solution to the clock angle problem today.

Problem:
When an analog clock reads 5:45, what is the radian measure of the angle between the hour hand and the minute hand?

Solution:

At 5:45, the minute hand will be exactly at the \(9\). In addition, since \(\dfrac34\) of an hour has passed, the hour hand will have moved \(\dfrac34\) of the way from the \(5\) to the \(6\), as in the diagram below.

Now, the angle between the \(6\) and the \(9\) (in blue) is \(\dfrac14\) of a circle, and the angle between the \(6\) and the hour hand (in purple) is \(\dfrac14\) of \(\dfrac1{12}\) of a circle, which equals \(\dfrac1{48}\). If we add these fractions, we find that the total shaded angle is \[\dfrac14+\dfrac1{48}=\dfrac{12}{48}+\dfrac1{48}=\dfrac{13}{48}.\]Since a full circle is \(2\pi\) radians, we can multiply this fraction by \(2\pi\), thus giving us an angle of \(\dfrac{13}{48}(2\pi)=\dfrac{13\pi}{24}\) radians. \(\blacksquare\)

Notice that at no point did we need to use degrees! In order to "speak radians fluently," you need to remember the key point from the previous reading:

Radians are angles measured as fractions of \(2\pi\).

Remembering this key piece of intuition is what will help you become "fluent" in radians as we go throughout the semester.

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Posted by Bill Shillito