if-cos-4-sin-4-2-5cos-then-find-the-value-of-

Question Number 159787 by abdullah_ff last updated on 21/Nov/21

if cos^4 θ − sin^4 θ = 2 − 5cosθ then find the value of θ

$$\mathrm{if}\:{cos}^{\mathrm{4}} \theta\:−\:{sin}^{\mathrm{4}} \theta\:=\:\mathrm{2}\:−\:\mathrm{5}{cos}\theta \\ $$$$\mathrm{then}\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\theta \\ $$

Commented by cortano last updated on 21/Nov/21

⇒ cos^2 θ−sin^2 θ−2+5cos θ = 0 ⇒cos^2 θ−1+cos^2 θ−2+5cos θ= 0 ⇒2cos^2 θ+5cos θ−3 = 0 ⇒(2cos θ−1)(cos θ+3)=0 ⇒cos θ=(1/2) ⇒θ=± (π/3)+2nπ

$$\Rightarrow\:\mathrm{cos}\:^{\mathrm{2}} \theta−\mathrm{sin}\:^{\mathrm{2}} \theta−\mathrm{2}+\mathrm{5cos}\:\theta\:=\:\mathrm{0} \\ $$$$\Rightarrow\mathrm{cos}\:^{\mathrm{2}} \theta−\mathrm{1}+\mathrm{cos}\:^{\mathrm{2}} \theta−\mathrm{2}+\mathrm{5cos}\:\theta=\:\mathrm{0}\: \\ $$$$\Rightarrow\mathrm{2cos}\:^{\mathrm{2}} \theta+\mathrm{5cos}\:\theta−\mathrm{3}\:=\:\mathrm{0} \\ $$$$\Rightarrow\left(\mathrm{2cos}\:\theta−\mathrm{1}\right)\left(\mathrm{cos}\:\theta+\mathrm{3}\right)=\mathrm{0} \\ $$$$\Rightarrow\mathrm{cos}\:\theta=\frac{\mathrm{1}}{\mathrm{2}}\:\Rightarrow\theta=\pm\:\frac{\pi}{\mathrm{3}}+\mathrm{2}{n}\pi \\ $$

Commented by abdullah_ff last updated on 23/Nov/21

$${thank}\:{you}\:{sir} \\ $$

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