Question Number 198604 by ajfour last updated on 22/Oct/23
Commented by ajfour last updated on 22/Oct/23
$$\theta+\phi=\alpha\:\left({known}\right) \\ $$$${radius}\:{of}\:{arc}\:{is}\:{unity}. \\ $$$${If}\:{the}\:{two}\:{shaded}\:{parts}\:{are}\:{equal}, \\ $$$${find}\:\theta={f}\left(\alpha\right). \\ $$
Answered by mr W last updated on 22/Oct/23
$$\frac{\alpha{R}^{\mathrm{2}} }{\mathrm{2}}=\frac{{R}\:\mathrm{cos}\:\theta×\left({R}\:\mathrm{cos}\:\theta\right)\:\mathrm{tan}\:\alpha}{\mathrm{2}} \\ $$$$\mathrm{cos}^{\mathrm{2}} \:\theta=\frac{\alpha}{\mathrm{tan}\:\alpha} \\ $$$$\Rightarrow\theta=\mathrm{cos}^{−\mathrm{1}} \sqrt{\frac{\alpha}{\mathrm{tan}\:\alpha}} \\ $$$${example}:\:\alpha=\mathrm{45}°,\:\Rightarrow\theta\approx\mathrm{27}.\mathrm{6}° \\ $$
Commented by ajfour last updated on 22/Oct/23
$${Thank}\:{you}!\:{sir}. \\ $$$$\alpha=\frac{\pi}{\mathrm{4}}\:\:\:\Rightarrow\:\:\mathrm{cos}\:^{\mathrm{2}} \theta=\frac{\pi}{\mathrm{4}}\:\:\:\:\:{as}\:\mathrm{tan}\:\frac{\pi}{\mathrm{4}}=\mathrm{1} \\ $$