Question and Answers Forum

All Questions      Topic List

Trigonometry Questions

Previous in All Question      Next in All Question      

Previous in Trigonometry      Next in Trigonometry      

Question Number 106468 by bemath last updated on 05/Aug/20

4cos x cos 2x cos 3x = 1

$$\mathrm{4cos}\:\mathrm{x}\:\mathrm{cos}\:\mathrm{2x}\:\mathrm{cos}\:\mathrm{3x}\:=\:\mathrm{1} \\ $$

Answered by john santu last updated on 05/Aug/20

recall : 2cos 3x cos x = cos 4x+cos 2x  ⇒2{cos 4x+cos 2x}cos 2x=1  2{2cos^2 2x−1+cos 2x}cos 2x=1  let cos 2x = ϑ   ⇒2(2ϑ^2 +ϑ−1)ϑ−1=0  4ϑ^3  + 2ϑ^2 −2ϑ−1 = 0  (2ϑ+1)(2ϑ^2 −1) = 0  case(1) cos 2x = −(1/2)   2x = ± ((2π)/3)+k.360° ; x=±(π/3)+k.180°  case(2) cos 2x = ((√2)/2)   2x = ± (π/4)+k.360° ; x= ±(π/8)+k.180°  case(3) cos 2x = −((√2)/2)  2x = ± ((3π)/4)+k.360° ; x= ±((3π)/8)+k.180°

$$\mathrm{recall}\::\:\mathrm{2cos}\:\mathrm{3x}\:\mathrm{cos}\:\mathrm{x}\:=\:\mathrm{cos}\:\mathrm{4x}+\mathrm{cos}\:\mathrm{2x} \\ $$$$\Rightarrow\mathrm{2}\left\{\mathrm{cos}\:\mathrm{4x}+\mathrm{cos}\:\mathrm{2x}\right\}\mathrm{cos}\:\mathrm{2x}=\mathrm{1} \\ $$$$\mathrm{2}\left\{\mathrm{2cos}\:^{\mathrm{2}} \mathrm{2x}−\mathrm{1}+\mathrm{cos}\:\mathrm{2x}\right\}\mathrm{cos}\:\mathrm{2x}=\mathrm{1} \\ $$$$\mathrm{let}\:\mathrm{cos}\:\mathrm{2x}\:=\:\vartheta\: \\ $$$$\Rightarrow\mathrm{2}\left(\mathrm{2}\vartheta^{\mathrm{2}} +\vartheta−\mathrm{1}\right)\vartheta−\mathrm{1}=\mathrm{0} \\ $$$$\mathrm{4}\vartheta^{\mathrm{3}} \:+\:\mathrm{2}\vartheta^{\mathrm{2}} −\mathrm{2}\vartheta−\mathrm{1}\:=\:\mathrm{0} \\ $$$$\left(\mathrm{2}\vartheta+\mathrm{1}\right)\left(\mathrm{2}\vartheta^{\mathrm{2}} −\mathrm{1}\right)\:=\:\mathrm{0} \\ $$$$\mathrm{case}\left(\mathrm{1}\right)\:\mathrm{cos}\:\mathrm{2x}\:=\:−\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\:\mathrm{2x}\:=\:\pm\:\frac{\mathrm{2}\pi}{\mathrm{3}}+\mathrm{k}.\mathrm{360}°\:;\:\mathrm{x}=\pm\frac{\pi}{\mathrm{3}}+\mathrm{k}.\mathrm{180}° \\ $$$$\mathrm{case}\left(\mathrm{2}\right)\:\mathrm{cos}\:\mathrm{2x}\:=\:\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\: \\ $$$$\mathrm{2x}\:=\:\pm\:\frac{\pi}{\mathrm{4}}+\mathrm{k}.\mathrm{360}°\:;\:\mathrm{x}=\:\pm\frac{\pi}{\mathrm{8}}+\mathrm{k}.\mathrm{180}° \\ $$$$\mathrm{case}\left(\mathrm{3}\right)\:\mathrm{cos}\:\mathrm{2x}\:=\:−\frac{\sqrt{\mathrm{2}}}{\mathrm{2}} \\ $$$$\mathrm{2x}\:=\:\pm\:\frac{\mathrm{3}\pi}{\mathrm{4}}+\mathrm{k}.\mathrm{360}°\:;\:\mathrm{x}=\:\pm\frac{\mathrm{3}\pi}{\mathrm{8}}+\mathrm{k}.\mathrm{180}° \\ $$

Commented by bemath last updated on 05/Aug/20

thank you

$$\mathrm{thank}\:\mathrm{you} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com