The Physics of limθ→(sinθ)/θ=1

Lovatt, Ian

doi:10.1119/1.5131126

The Physics Teacher

2019

DOI: 10.1119/1.5131126

|View full text |Cite

The Physics of limθ→(sinθ)/θ=1

Ian Lovatt

Abstract: Students in first-year calculus meet limθ→0(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…

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...

Citation Types

Supporting

Mentioning

Contrasting

Year Published

2019

2022

Publication Types

Select...

Article2

Relationship

Self Cite0

Independent2

Authors

Journals

Cited by 2 publications

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

On the dimension of angles and their units

et al. 2022

View full text Add to dashboard Cite

We show the implications of angles having their own dimension, which facilitates a consistent use of units as is done for lengths, masses, and other physical quantities. We do this by examining the properties of complete trigonometric and exponential functions that are generalizations of the corresponding functions that have dimensionless numbers for arguments. These generalizations provide functions that are independent of units in which the angles are expressed. This property also provides a consistent framework for including quantities involving angles in computer algebra programs without ambiguity that may otherwise occur. This is in contrast to the conventional practice in scientific applications involving trigonometric or exponential functions of angles where it is assumed that the argument is the numerical part of the angle when expressed in units of radians. That practice also assumes that the functions are the corresponding radian-based versions. These assumptions allow angles to be treated as if they had no dimension and no units, an approach that can lead to important difficulties such as incorrect factors of $2\pi$, which can be avoided by assigning an independent dimension to angles.

show abstract