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The Derivative of Versed Sine


Versine

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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.


Sources:

  1. FooPlot: Online graphing calculator and function plotter

Recommended:

Category: Math Calculus Trigonometry


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This is a student's notebook. I am not responsible if you copy it for homework, and it turns out to be wrong.

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August 05, 2017

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