Approximating θ/2 as θ for small angles: Is the error significant?

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SUMMARY

The discussion centers on the approximation of θ/2 as θ for small angles, highlighting that while both angles are of the order O(θ), the relative error remains O(1). The key mathematical expression presented is the relative error calculation, which shows that the difference between θ/2 and θ, when normalized by θ/2, equals 1. This indicates that the approximation may not be suitable for precise calculations where accuracy is critical.

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physics user1
θ/2 as θ if θ is a small angle?

Thanks
 
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That depends on how you quantify the error. For example, both angles are ##O(\theta)## of course, but the relative error is still ##O(1)##. By the latter I mean that
$$
\frac{\left|\frac{\theta}{2} - \theta \right|}{\left|\frac{\theta}{2}\right|} = 1
$$
 
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