$$\definecolor{flatred}{RGB}{192, 57, 43} \definecolor{flatblue}{RGB}{41, 128, 185} \definecolor{flatgreen}{RGB}{39, 174, 96} \definecolor{flatpurple}{RGB}{142, 68, 173} \definecolor{flatlightblue}{RGB}{52, 152, 219} \definecolor{flatlightred}{RGB}{231, 76, 60} \definecolor{flatlightgreen}{RGB}{46, 204, 113} \definecolor{flatteal}{RGB}{22, 160, 133} \definecolor{flatpink}{RGB}{243, 104, 224}$$

February 12, 2021

Clock angles solution

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.