Question Number 20506 by Tinkutara last updated on 27/Aug/17
$${Simplify}: \\ $$$$\mathrm{cos}^{−\mathrm{1}} \:\left(\frac{\mathrm{sin}\:{x}\:+\:\mathrm{cos}\:{x}}{\:\sqrt{\mathrm{2}}}\right),\:\frac{\pi}{\mathrm{4}}\:<\:{x}\:<\:\frac{\mathrm{5}\pi}{\mathrm{4}} \\ $$
Answered by ajfour last updated on 27/Aug/17
![θ=cos^(−1) [cos (π/4)cos x+sin (π/4)sin x] =cos^(−1) [cos (x−π/4)] And as π/4 < x < 5π/4 so 0 < x−π/4 < π Hence 𝛉= x−𝛑/4 .](https://www.tinkutara.com/question/Q20514.png)
$$\:\theta=\mathrm{cos}^{−\mathrm{1}} \left[\mathrm{cos}\:\left(\pi/\mathrm{4}\right)\mathrm{cos}\:{x}+\mathrm{sin}\:\left(\pi/\mathrm{4}\right)\mathrm{sin}\:{x}\right] \\ $$$$\:\:=\mathrm{cos}^{−\mathrm{1}} \left[\mathrm{cos}\:\left({x}−\pi/\mathrm{4}\right)\right] \\ $$$${And}\:{as}\:\:\:\:\:\pi/\mathrm{4}\:<\:{x}\:<\:\mathrm{5}\pi/\mathrm{4} \\ $$$${so}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{0}\:<\:{x}−\pi/\mathrm{4}\:<\:\pi \\ $$$${Hence}\:\:\boldsymbol{\theta}=\:\boldsymbol{{x}}−\boldsymbol{\pi}/\mathrm{4}\:. \\ $$
Commented by Tinkutara last updated on 27/Aug/17
$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{very}\:\mathrm{much}\:\mathrm{Sir}! \\ $$