Question Number 123018 by ajfour last updated on 21/Nov/20
Commented by ajfour last updated on 21/Nov/20
$${If}\:{rectangles}\:{ABCD}\:{and}\:{PQRS} \\ $$$${are}\:{congruent},\:{find}\:\theta,\:{in}\:{terms}\:{of}\:\boldsymbol{\alpha}. \\ $$
Commented by mr W last updated on 22/Nov/20
$${AB}={a} \\ $$$${AF}={b}={a}\:\mathrm{tan}\:\alpha \\ $$$${b}={AQ}+{QD}={a}\:\mathrm{tan}\:\theta+{a}\:\mathrm{cos}\:\theta={a}\:\mathrm{tan}\:\alpha \\ $$$$\mathrm{tan}\:\theta+\mathrm{cos}\:\theta=\mathrm{tan}\:\alpha \\ $$$${let}\:\lambda=\mathrm{tan}\:\alpha \\ $$$$\mathrm{1}+\left(\mathrm{1}+\lambda^{\mathrm{2}} \right)\:\mathrm{cos}^{\mathrm{2}} \:\theta−\mathrm{2}\lambda\:\mathrm{cos}^{\mathrm{3}} \:\theta+\mathrm{cos}^{\mathrm{4}} \:\theta=\mathrm{0} \\ $$$$\Rightarrow\mathrm{cos}\:\theta=... \\ $$