Small-angle approximation: Difference between revisions

Revision as of 09:01, 7 March 2025

Revision as of 17:51, 7 March 2025

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== Justifications ==

=== Graphic ===

The accuracy of the approximations can be seen below in Figure 1 and Figure 2. As the measure of the angle approaches zero, the difference between the approximation and the original function also approaches 0.

<gallery widths="300px" heights="240px">

File:Small_angle_compair_odd.svg|'''Figure 1.''' A comparison of the basic [[odd function|odd]] trigonometric functions to {{mvar|θ}}. It is seen that as the angle approaches 0 the approximations become better.

File:Small_angle_compare_even.svg|'''Figure 2.''' A comparison of {{math|cos ''θ''}} to {{math|1 − {{sfrac|''θ''²|2}}}}. It is seen that as the angle approaches 0 the approximation becomes better.

</gallery>

=== Geometric ===

[[File:Small angle triangle.svg|600px400px]]

The red section on the right, {{math|d}}, is the difference between the lengths of the hypotenuse, {{mvar|H}}, and the adjacent side, {{mvar|A}}. As is shown, {{mvar|H}} and {{mvar|A}} are almost the same length, meaning {{math|cos ''θ''}} is close to 1 and {{math|{{sfrac|''θ''²|2}}}} helps trim the red away.

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

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==Error of the approximations==

[[File:Small angle compare error.svg|thumb|300px|'''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.