Small-angle approximation: Difference between revisions

Revision as of 05:53, 8 March 2025

Revision as of 08:27, 8 March 2025

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[[File:Small angle triangle.svg|frameless|upright=2]]

For a small angle, {{mvar|H}} and {{mvar|A}} are almost the same length, and therefore {{math|cos ''θ''}} is nearly 1. The segment {{mvar|d}} (in red to the right) is the difference between the lengths of the hypotenuse, {{mvar|H}}, and the adjacent side, {{mvar|A}}, and has length <math>\textstyle H - \sqrt{H^2 - O^2}</math>, which for small angles is approximately equal to <math>\textstyle O^2\!/2H \approx \tfrac12 \theta^2H</math>. As a second -order approximation,

<math display="block"> \cos{\theta} \approx 1 - \frac{\theta^2}{2}.</math>