Small-angle approximation: Difference between revisions

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=== Algebraic ===

[[File:Small-angle approximation for sine function.svg|thumb|300px|The small-angle approximation for the sine function.]]

The Maclaurin[[Trigonometric expansionfunctions#Power (theseries expansion|Taylor expansionseries about 0)expansions of thetrigonometric relevantfunctions]] trigonometricsine, cosine, and tangent near functionzero isare:<ref>{{cite book| authorlink=Mary L. Boas|last=Boas| first=Mary L.|title=[[Mathematical Methods in the Physical Sciences]]|year=2006| publisher=Wiley|page=26| isbn=978-0-471-19826-0}}</ref>

<math display="block">\begin{align}

\sin \theta &= \sum^{\infin}_{n=0} \frac{(-1)^n}{(2n+1)!} \theta^{2n+1} \\

\sin \theta &= \theta - \frac{frac16\theta^3}{3!} + \fracfrac1{120}\theta^5}{5!} - \frac{\theta^7}{7!} +cdots, \cdots\[6mu]

\cos \theta &= 1 - \frac1{2}{\theta^2} + \frac1{24}\theta^4 - \cdots, \\[6mu]

<math display="block">\sintan \theta &= \theta -+ \frac{1}{3}\theta^3}{6} + \frac{\theta^52}{12015} - \frac{\theta^7}{5040}5 + \cdots </math>.

\end{align}</math>

where {{mvar|θ}} is the angle in radians. In clearer terms,

<math display="block">\sin \theta = \theta - \frac{\theta^3}{6} + \frac{\theta^5}{120} - \frac{\theta^7}{5040} + \cdots </math>

where {{tmath|\theta}} is the angle in radians. For very small angles, higher powers of {{tmath|\theta}} become extremely small, for instance if {{tmath|1= \theta = 0.01}}, then {{tmath|1= \theta^3 = 0.000\,001}}, just one ten-thousandth of {{tmath|\theta}}. Thus for many purposes it is safe to drop the cubic and higher terms and approximate the sine and tangent as

It is readily seen that the second most significant (third-order) term falls off as the cube of the first term; thus, even for a not-so-small argument such as 0.01, the value of the second most significant term is on the order of {{val|0.000001}}, or {{sfrac|{{val|10000}}}} the first term. One can thus safely approximate:

<math display="block">\sin\theta \approx \tan\theta \approx \theta.</math>

The cosine of a sufficiently small angle is very nearly {{tmath|1}}, which sometimes suffices as an approximation. Where more precision is needed the quadratic term can also included, and cosine can be approximated as

<math display="block">\tan cos\theta \approx \sin1 \theta- \approx tfrac12\theta,^2.</math>

By extension, since the cosine of a small angle is very nearly 1, and the tangent is given by the sine divided by the cosine,

<math display="block">\tan \theta \approx \sin \theta \approx \theta,</math>

==== Dual numbers ====