Question

If the points (a cosα, a sinα) and (a cosβ, a sinβ) are at a distance k $\sin \left(\frac{\alpha -\beta }{2}\right)$ apart, then k = __________.

Answer 1

Given (a cos α, a sin α) and (a cos β, a sin β) are at a distance k $\sin \left(\frac{\alpha -\beta }{2}\right)$ a part
Using distance formula,

$$\begin{gathered} \sqrt{{(a\cos \beta -a\cos \alpha )}^2+{(a\sin \beta -a\sin \alpha )}^2} \\ =\sqrt{a^2{\left(\cos \beta -\cos \alpha \right)}^2+a^2{\left(\sin \beta -\sin \alpha \right)}^2} \\ =a\sqrt{{\cos }^2\beta +{\cos }^2\alpha -2\cos \alpha \cos \beta +{\sin }^2\alpha +{\sin }^2\beta -2\sin \alpha \sin \beta } \\ =a\sqrt{1+1-2\cos \alpha \cos \beta -2\sin \alpha \sin \beta } \\ =a\sqrt{2\left(1-\left(\cos \alpha \cos \beta -\sin \alpha \sin \beta \right)\right)} \\ =\sqrt{2}a\sqrt{1-\cos \left(\alpha -\beta \right)} \\ =\sqrt{2}a\sqrt{2{\sin }^2\left(\frac{\alpha -\beta }{2}\right)} \\ =2a\sin \left(\frac{\alpha -\beta }{2}\right) \\ \mathrm{i}.\mathrm{e}K=2a \end{gathered}$$