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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
...
...
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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| Excerpt |
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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} |
If the angle is small enough, then we can ignore all but the first term, giving
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{latex} \\ If the angle is small enough, then 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
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will
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be
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the
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case
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when
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x
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is
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much
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greater
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than
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x
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3
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/3!
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,
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or
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in
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other
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words
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6
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>>
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x
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2
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.
Similarly,
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the
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Taylor
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series
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for
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cos(
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θ) is
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* is \\ {latex}\begin{large} \[ cos( \theta ) = 1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \frac({x^{6}}{56!} + ... \]\end{large}{latex} \\ |
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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{latex}\begin{large} \[ costan( \theta ) = \theta + \frac{x^{3}}{3} + \frac{2x^{5}}{15} \approx 1+ \frac{17x^{7}}{217} + ... \]\end{large} |
So that for small angles
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\begin{large} \[ tan( \theta ) \approx \theta \]\end{large}{latex} \\ |