Students in first-year calculus meet limθ0(sinθ)/θ=1, perhaps as a step toward finding the derivative of sin θ. If they see a proof, it involves a version of the so-called squeeze theorem, resulting in cos θ ≤(sin θ)/ θ ≤ 1/(cos θ). All very nice, rigorous, elegant even, but…

1.
Martin
Gardner
, “
Mathematical games
,”
Sci. Am.
229
(
4
),
114
(
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1973
).
2.
Eli
Maor
, “
A remarkable trigonometric identity
,”
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452
(
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3.
In the appendix, I have given an example of the squeeze theorem. The appendix can be viewed at TPT Online under the Supplemental tab, http://dx.doi.org/10.1119/1.5131126.
Claudi
Alsina
and
Roger B.
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Teaching tip: The limit of (sin t)/t
,”
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,” Better Explained website, https://betterexplained.com/articles/intuitive-guide-to-angles-degrees-and-radians/.
5.
For example, see
William
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.
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,”
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Jean-Marc
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,”
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8.
The AAPT Metric Education and SI Practice Committee recommends that “the radian should explicitly appear…when different numerical values for a quantity would and should be obtained if other angle measures were used.”
Gordon J.
Aubrecht
 II
 et al, “
The radian – That troublesome unit
,”
Phys. Teach.
31
,
84
(
Feb.
1993
). In the case of the area-density introduced in this paper, the number of square meters per angle does depend on the unit used to measure angle.
9.
The expression for normalized density is valid only for angles measured in radians. Similarly, the expression for tangential speed vt (vt = ) is valid only when angles are measured in radians. The Metric Education Committee (Ref. 8) recommends that the radian be expressed in units of m/m; this paper’s normalized density is thus dimensionless—just a ratio of two densities.

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