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Question Number 199968 by a.lgnaoui last updated on 11/Nov/23

determiner x ?

$$\mathrm{determiner}\:\boldsymbol{\mathrm{x}}\:\:? \\ $$

Commented by a.lgnaoui last updated on 11/Nov/23

Answered by mr W last updated on 12/Nov/23

Commented by mr W last updated on 12/Nov/23

let x=α, r=radius γ=2β+α=90°−α ⇒β=45°−α tan β=(r/(r+(r/(tan α))))=(1/(1+(1/(tan α))))=((1−tan α)/(1+tan α)) ⇒tan α=(1/2) ⇒α=x=tan^(−1) (1/2)≈26.565°

$${let}\:{x}=\alpha,\:{r}={radius} \\ $$$$\gamma=\mathrm{2}\beta+\alpha=\mathrm{90}°−\alpha \\ $$$$\Rightarrow\beta=\mathrm{45}°−\alpha \\ $$$$\mathrm{tan}\:\beta=\frac{{r}}{{r}+\frac{{r}}{\mathrm{tan}\:\alpha}}=\frac{\mathrm{1}}{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{tan}\:\alpha}}=\frac{\mathrm{1}−\mathrm{tan}\:\alpha}{\mathrm{1}+\mathrm{tan}\:\alpha} \\ $$$$\Rightarrow\mathrm{tan}\:\alpha=\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\Rightarrow\alpha={x}=\mathrm{tan}^{−\mathrm{1}} \frac{\mathrm{1}}{\mathrm{2}}\approx\mathrm{26}.\mathrm{565}° \\ $$

Commented by a.lgnaoui last updated on 12/Nov/23

thanks

$$\mathrm{thanks} \\ $$

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