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A very useful approximation for many physical applications, especially for [simple harmonic motion] and [pendulums|pendulum] in particular. It states that {excerpt} when the angle is small, and expressed in [radians], then we may approximate *sin(θ)* 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*:
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{latex}\begin{large} \[ sin( \theta ) = x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \frac{x^{7}}{7!} + ... \]\end{large}{latex}

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If the angle is small enough, then we can ignore all but the first term, giving

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{latex}\begin{large} \[ sin( \theta ) \approx x\theta \]\end{large}{latex}

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This gives us our criterion for "small enough", because it is clear that this will be the case when *x* is much greater than {*}x{^}3{^}/3!{*}, or in other words *6 >> x{^}2{^}*.


Similarly, the Taylor series for *cos(θ)* is

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{latex}\begin{large} \[ cos( \theta ) = 1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \frac{x^{6}}{6!} + ... \]\end{large}{latex}

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so that in the small angle approximation we have 
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{latex}\begin{large} \[ cos( \theta ) \approx 1 \]\end{large}{latex}

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Finally, the expansion for the *tangent* is
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{latex}\begin{large} \[ tan( \theta ) = \theta + \frac{x^{3}}{3} + \frac{2x^{5}}{15} + \frac{17x^{7}}{217} + ... \]\end{large}{latex}
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So that for small angles 
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{latex}\begin{large} \[ tan( \theta ) \approx \theta \]\end{large}{latex}
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