Menu Close

cosx-cos3x-cos5x-cos7x-x-




Question Number 207518 by hardmath last updated on 17/May/24
cosx cos3x  =  cos5x cos7x  ⇒ x = ?
$$\mathrm{cos}\boldsymbol{\mathrm{x}}\:\mathrm{cos3}\boldsymbol{\mathrm{x}}\:\:=\:\:\mathrm{cos5}\boldsymbol{\mathrm{x}}\:\mathrm{cos7}\boldsymbol{\mathrm{x}} \\ $$$$\Rightarrow\:\boldsymbol{\mathrm{x}}\:=\:? \\ $$
Answered by Frix last updated on 17/May/24
cos x cos 3x =cos 5x cos 7x  ((cos 2x +cos 4x)/2)=((cos 2x +cos 12x)/2)  cos 4x =cos 12x  cos y =cos 3y  cos y =(1−4sin^2  y)cos y  cos y −(1−4sin^2  y)cos y =0  4cos y sin^2  y =0  cos y =0 ∨sin y =0  cos 4x =0 ∨sin 4x =0  x=(((2n+1)π)/8)∨x=((2nπ)/8)  x=((nπ)/8)
$$\mathrm{cos}\:{x}\:\mathrm{cos}\:\mathrm{3}{x}\:=\mathrm{cos}\:\mathrm{5}{x}\:\mathrm{cos}\:\mathrm{7}{x} \\ $$$$\frac{\mathrm{cos}\:\mathrm{2}{x}\:+\mathrm{cos}\:\mathrm{4}{x}}{\mathrm{2}}=\frac{\mathrm{cos}\:\mathrm{2}{x}\:+\mathrm{cos}\:\mathrm{12}{x}}{\mathrm{2}} \\ $$$$\mathrm{cos}\:\mathrm{4}{x}\:=\mathrm{cos}\:\mathrm{12}{x} \\ $$$$\mathrm{cos}\:{y}\:=\mathrm{cos}\:\mathrm{3}{y} \\ $$$$\mathrm{cos}\:{y}\:=\left(\mathrm{1}−\mathrm{4sin}^{\mathrm{2}} \:{y}\right)\mathrm{cos}\:{y} \\ $$$$\mathrm{cos}\:{y}\:−\left(\mathrm{1}−\mathrm{4sin}^{\mathrm{2}} \:{y}\right)\mathrm{cos}\:{y}\:=\mathrm{0} \\ $$$$\mathrm{4cos}\:{y}\:\mathrm{sin}^{\mathrm{2}} \:{y}\:=\mathrm{0} \\ $$$$\mathrm{cos}\:{y}\:=\mathrm{0}\:\vee\mathrm{sin}\:{y}\:=\mathrm{0} \\ $$$$\mathrm{cos}\:\mathrm{4}{x}\:=\mathrm{0}\:\vee\mathrm{sin}\:\mathrm{4}{x}\:=\mathrm{0} \\ $$$${x}=\frac{\left(\mathrm{2}{n}+\mathrm{1}\right)\pi}{\mathrm{8}}\vee{x}=\frac{\mathrm{2}{n}\pi}{\mathrm{8}} \\ $$$${x}=\frac{{n}\pi}{\mathrm{8}} \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *