Cullen Banks

2023-03-14

What is $\mathrm{csc}\left(\frac{\theta }{2}\right)$ in terms of trigonometric functions of a unit $\theta$?

Edwin Simmons

Recall $\mathrm{cos}2A=1-2{\mathrm{sin}}^{2}A$
i.e. $2{\mathrm{sin}}^{2}A=1-\mathrm{cos}2A$
or ${\mathrm{sin}}^{2}A=\frac{1-\mathrm{cos}2A}{2}$
or $\mathrm{sin}A=\sqrt{\frac{1-\mathrm{cos}2A}{2}}$
Putting $2A=\theta$ or $A=\frac{\theta }{2}$
Therefore, $\mathrm{sin}\left(\frac{\theta }{2}\right)=\sqrt{\frac{1-\mathrm{cos}\theta }{2}}$
or $\mathrm{csc}\left(\frac{\theta }{2}\right)=\sqrt{\frac{2}{1-\mathrm{cos}\theta }}$

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