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pendular motion is also already discussed |
mention that angles measured in degrees must be scaled |
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:<math>
\begin{align}
\sin \theta &\approx \tan \theta \approx \theta, \\[5mu]
\cos \theta &\approx 1 - \
\end{align}
</math>
Angles measured in [[degree (angle)|degrees]] must first be converted to radians by multiplying them by {{tmath|\pi/180}}.
These approximations have a wide range of uses in branches of [[physics]] and [[engineering]], including [[mechanics]], [[electromagnetics|electromagnetism]], [[optics]], [[cartography]], [[astronomy]], and [[computer science]].<ref name="Holbrow2010" /><ref name="Plesha2012"/> One reason for this is that they can greatly simplify [[differential equation]]s that do not need to be answered with absolute precision.
There are a number of ways to demonstrate the validity of the small-angle approximations. The most direct method is to truncate the [[Maclaurin series]] for each of the trigonometric functions. Depending on the [[Order of approximation#Usage_in_science_and_engineering|order of the approximation]], <math>\textstyle \cos \theta</math> is approximated as either <math>1</math> or as <math display=
== Justifications ==
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==== Dual numbers ====
One may also use [[dual numbers]], defined as numbers in the form <math>a + b\varepsilon</math>, with <math>a,b\in\mathbb R</math> and <math>\varepsilon</math> satisfying by definition <math>\varepsilon^2 = 0</math> and <math>\varepsilon \ne 0</math>. By using the MacLaurin series of cosine and sine, one can show that <math>\cos(\theta\varepsilon) = 1</math> and <math>\sin(\theta\varepsilon) = \theta\varepsilon</math>. Furthermore, it is not hard to prove that the [[Pythagorean identity]] holds:<math display="block">\sin^2(\theta\varepsilon) + \cos^2(\theta\varepsilon) = (\theta\varepsilon)^2 + 1^2 = \theta^2\varepsilon^2 + 1 = \theta^2 \cdot 0 + 1 = 1</math>
==Error of the approximations==
[[File:Small angle compare error.svg|thumb|upright=2|'''Figure 3.''' A graph of the [[relative error]]s for the small angle approximations.]]
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* {{math|sin ''θ'' ≈ ''θ''}} at about 0.2441 radians (13.99°)
* {{math|cos ''θ'' ≈ 1 − {{sfrac|''θ''<sup>2</sup>|2}}}} at about 0.6620 radians (37.93°)
== Angle sum and difference ==
The [[angle addition and subtraction theorems]] reduce to the following when one of the angles is small (''β'' ≈ 0):
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