Small-angle approximation: Difference between revisions

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{{tmath|\cos\theta \approx 1 - \tfrac12\theta^2}}, and

{{tmath|\tan\theta \approx \theta + \tfrac13\theta^3}}.

==== 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==

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=== Astronomy ===

In [[astronomy]], the [[angular size]] or angle subtended by the image of a distant object is often only a few [[arcsecond]]s (denoted by the symbol ″), so it is well suited to the small angle approximation.<ref name=Green1985/> The linear size ({{mvar|D}}) is related to the angular size ({{mvar|X}}) and the distance from the observer ({{mvar|d}}) by the simple formula:

:<math>D = X \frac{d}{206\,265{''}}</math>

where {{mvar|X}} is measured in arcseconds.

The quantity {{val|206265|u="}} is approximately equal to the number of arcseconds in 1 radian, which is the number of arcseconds in a [[circle]] ({{val|1296000|u="}}), divided by {{math|2π}}, or, the number of arcseconds in 1 radian.

The exact formula is

:<math>D = d \tan \left( X \frac{2\pi}{1\,296\,000{''}} \right)</math>

and the above approximation follows when {{math|tan ''X''}} is replaced by {{mvar|X}}.