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A
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very
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useful
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approximation
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for
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many
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physical
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applications,
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especially
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for
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and
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in
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particular.
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It
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states
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that
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when |
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the |
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angle |
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is |
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small, |
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and |
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expressed |
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in |
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, |
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then |
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we |
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may |
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approximate |
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sin( |
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θ) by θ. |
This follows because the sine function may be expressed as the infinite Taylor series:
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* by {*}θ{*}.{excerpt} At the same time, we may approximate *cos(θ)* by *1* and *tan(θ)* by {*}θ{*}. This follows because the sine function may be expressed as the infinite *Taylor series*: \\ {latex}\begin{large} \[ sin( \theta ) = x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \frac{x^{7}}{7!} + ... \]\end{large}{latex} \\ If the angle is small enough, then |
If the angle is small enough, then we can ignore all but the first term, giving
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we can ignore all but the first term, giving \\ {latex}\begin{large} \[ sin( \theta ) \approx x\theta \]\end{large}{latex} \\ This gives us our criterion for "small |
This gives us our criterion for "small enough",
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because
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it
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is
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clear
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that
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this will be the case when x is much greater than x3/3!, or in other words 6 >> x2.
Similarly, the Taylor series for cos(θ) is
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\begin{large} \[ cos( \theta ) = 1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \frac{x^{6}}{6!} + ... \]\end{large} |
so that in the small angle approximation we have
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\begin{large} \[ cos( \theta ) \approx 1 \]\end{large} |
Finally, the expansion for the tangent is
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\begin{large} \[ tan( \theta ) = \theta + \frac{x^{3}}{3} + \frac{2x^{5}}{15} + \frac{17x^{7}}{217} + ... \]\end{large} |
So that for small angles
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\begin{large} \[ tan( \theta ) \approx \theta \]\end{large} will be the case when *x* is much greater than {*}x{^}3{^}/3!{*}, or in other words *6 >> x{^}2{^}*. |