Question Number 41444 by Rio Michael last updated on 07/Aug/18
$${Given}\:{that}\:\:\:{cos}\left(\theta\:+\:\frac{\pi}{\mathrm{3}}\right)=\:\frac{\mathrm{1}}{\mathrm{2}}\:{find}\:{the}\:{values}\:{of}\:\theta\:{inthe}\:{range}\:\:\mathrm{0}°\leqslant\:\theta\leqslant\mathrm{360}° \\ $$
Commented by maxmathsup by imad last updated on 07/Aug/18
![cos(θ +(π/3)) =(1/2) ⇔ cos(θ +(π/3))=cos((π/3))⇔θ +(π/3) =(π/3) +2kπ or θ +(π/3)=−(π/3) +2kπ (k∈Z)⇔ θ=2kπ or θ =−((2π)/3) +2kπ now let sove in[0,2π] 0≤2kπ≤2π ⇒ k=0 or k=1 0≤−((2π)/3)+2kπ≤2π ⇒0≤−(2/3)+2k≤2 ⇒(2/3)≤2k≤2+(2/3) ⇒ (1/3)≤ k ≤ (4/3) ⇒ 0,...≤k≤1,.. ⇒ k=1 ⇒ x=2π −((2π)/3) =((4π)/3) the slutions are 0 , 2π and ((4π)/3)](Q41457.png)
$${cos}\left(\theta\:+\frac{\pi}{\mathrm{3}}\right)\:=\frac{\mathrm{1}}{\mathrm{2}}\:\Leftrightarrow\:{cos}\left(\theta\:+\frac{\pi}{\mathrm{3}}\right)={cos}\left(\frac{\pi}{\mathrm{3}}\right)\Leftrightarrow\theta\:+\frac{\pi}{\mathrm{3}}\:=\frac{\pi}{\mathrm{3}}\:+\mathrm{2}{k}\pi\:{or} \\ $$$$\theta\:+\frac{\pi}{\mathrm{3}}=−\frac{\pi}{\mathrm{3}}\:+\mathrm{2}{k}\pi\:\left({k}\in{Z}\right)\Leftrightarrow\:\:\theta=\mathrm{2}{k}\pi\:\:{or}\:\theta\:=−\frac{\mathrm{2}\pi}{\mathrm{3}}\:+\mathrm{2}{k}\pi\:\:{now}\:{let}\:{sove}\:{in}\left[\mathrm{0},\mathrm{2}\pi\right] \\ $$$$\mathrm{0}\leqslant\mathrm{2}{k}\pi\leqslant\mathrm{2}\pi\:\Rightarrow\:\:{k}=\mathrm{0}\:{or}\:{k}=\mathrm{1} \\ $$$$\mathrm{0}\leqslant−\frac{\mathrm{2}\pi}{\mathrm{3}}+\mathrm{2}{k}\pi\leqslant\mathrm{2}\pi\:\Rightarrow\mathrm{0}\leqslant−\frac{\mathrm{2}}{\mathrm{3}}+\mathrm{2}{k}\leqslant\mathrm{2}\:\Rightarrow\frac{\mathrm{2}}{\mathrm{3}}\leqslant\mathrm{2}{k}\leqslant\mathrm{2}+\frac{\mathrm{2}}{\mathrm{3}}\:\Rightarrow \\ $$$$\frac{\mathrm{1}}{\mathrm{3}}\leqslant\:{k}\:\:\leqslant\:\frac{\mathrm{4}}{\mathrm{3}}\:\Rightarrow\:\mathrm{0},...\leqslant{k}\leqslant\mathrm{1},..\:\Rightarrow\:{k}=\mathrm{1}\:\Rightarrow\:{x}=\mathrm{2}\pi\:−\frac{\mathrm{2}\pi}{\mathrm{3}}\:=\frac{\mathrm{4}\pi}{\mathrm{3}}\:\:{the}\:{slutions}\:{are} \\ $$$$\mathrm{0}\:,\:\mathrm{2}\pi\:\:{and}\:\frac{\mathrm{4}\pi}{\mathrm{3}} \\ $$