Small-angle approximation: Difference between revisions

Revision as of 09:13, 30 November 2024

Revision as of 17:40, 30 November 2024

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\end{align}</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 safesuffices to drop the cubic and higher terms and approximate the sine and tangent of a small angle using the radian measure of the angle, {{tmath|\sin\theta \approx \tan\theta \approx \theta}}, and drop the quadratic term and approximate the cosine as {{tmath|\cos\theta \approx 1}}.

If additional precision is needed the quadratic and cubic terms can also be included,

<math display=block>{{tmath|\sin\theta \approx \tan\theta - \approx tfrac16\theta.</math>^3}},

<math display=block>{{tmath|\cos\theta \approx 1 - \tfrac12\theta^2.</math>}}, and

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

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

<math display=block>\cos\theta \approx 1 - \tfrac12\theta^2.</math>

==== Dual numbers ====