Here’s a straightforward application of arctangent that, a generation ago, used to be taught in a typical Precalculus class (or, as it was called back then, analytical geometry).

Find the smallest angle between the lines and .

This problem is almost equivalent to finding the angle between the vectors and . I use the caveat *almost* because the angle between two vectors could be between and , while the smallest angle between two lines must lie between and .

This smallest angle can be found using the formula

,

where and are the slopes of the two lines. In the present case,

.

Not surprisingly, we obtain the same answer that we obtained a couple of posts ago using arccosine. The following picture makes clear why .

In tomorrow’s post, I’ll explain why the above formula actually works.

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