Small-angle approximation: Difference between revisions

Revision as of 02:25, 29 October 2006

Revision as of 02:25, 29 October 2006

Line 19:

The Taylor series of the trigonometric functions are

:<math>\sin\left( x \right) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + ...\cdots </math>

:<math>\cos\left( x \right) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + ...\cdots </math>

:<math>\tan\left( x \right) = x + \frac{x^3}{3} + \frac{2 x^5}{15} + \frac{17 x^7}{315} + ...\cdots</math>

When the angle ''x'' is less than one radian, its powers ''x''², ''x''³, ... decrease exponentially, so only a few are needed. The highest power included is called the order of the approximation. Neither sin(''x'') nor tan(''x'') has an ''x''² term, so their first- and second-order approximations are the same.

== Specific uses ==