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The versed sine ({$\operatorname{ver}$}) is equal to {$ 1 - \cos $} as can be seen by inspection of the unit circle diagram. It is the rest of the unit circle radius after cosine leaves off. As with many things in trigonometry there are a variety of ways to express the versed sine including: {$$ \operatorname{ver}\theta = 2\sin^2 \left( {\theta \over 2} \right) $$} which seems to be the preferred form in spite of not being particularly intuitive.

{$ y = \operatorname{arcver}(x) $} and {$ y=\operatorname{ver}(x) $}

{$$ \begin{align} \operatorname{ver}\theta &= 1 - \cos\theta \cr {d \over {d\theta}}\operatorname{ver}\theta &= {d \over {d\theta}} \left( 1 - \cos\theta \right) \cr &= {d \over {d\theta}} 1 - {d \over {d\theta}}\cos\theta \cr &= 0 - (-\sin\theta) \cr \therefore \quad {d \over {d\theta}}\operatorname{ver}\theta &= \sin\theta \end{align}$$}

Verify this, as this value is not given in many tables.

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**Category:** Math Calculus Trigonometry

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