Small-angle approximation: Difference between revisions

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==Error of the approximations==

[[File:Small angle compare error.svg|thumb|upright=2|'''Figure 3.''' A graph of the [[relative error]]s for the small angle approximations.]]

Near zero, the relative error of the approximations {{tmath|\cos \theta \approx 1}}, {{tmath|\sin \theta \approx \theta}}, and {{tmath|\tan \theta \approx \theta}} is quadratic in {{tmath|\theta}}: for each order of magnitude smaller the angle is, the relative error of these approximations shrinks by two orders of magnitude. The approximation {{tmath|\textstyle \cos \theta \approx 1 - \tfrac12\theta^2 }} has relative error which is quartic in {{tmath|\theta}}: for each order of magnitude smaller the angle is, the relative error shrinks by four orders of magnitude.

Figure 3 shows the relative errors of the small angle approximations. The angles at which the relative error exceeds 1% are as follows:

* {{mathtmath|\cos ''θ''\theta ≈\approx 1}} at about 0.140814 radians (8.071°)

* {{mathtmath|\tan ''θ''\theta ≈\approx ''θ''\theta}} at about 0.173017 radians (9.919°)

* {{mathtmath|\sin ''θ''\theta ≈\approx ''θ''\theta}} at about 0.244124 radians (1314.990°)

* {{mathtmath|\textstyle \cos ''θ''\theta ≈\approx 1 −- {{sfrac|''θ''^{\tfrac12\theta^2}|2}} }} at about 0.662066 radians (37.939°)

== Angle sum and difference ==