Menu Close

Question-176891




Question Number 176891 by Shrinava last updated on 27/Sep/22
Answered by mr W last updated on 27/Sep/22
sin x cos 3x (1/2)=2 sin x cos x (((cos x)/2)+(((√3) sin x)/2))(((cos x)/2)−(((√3) sin x)/2))  cos 3x = cos x (cos^2  x−3 sin^2  x)  4 cos^2  x−3 = cos^2  x−3 sin^2  x  3−3 =0  0=0 ✓  ⇒any x fulfills the equation and is  a solution.
$$\mathrm{sin}\:{x}\:\mathrm{cos}\:\mathrm{3}{x}\:\frac{\mathrm{1}}{\mathrm{2}}=\mathrm{2}\:\mathrm{sin}\:{x}\:\mathrm{cos}\:{x}\:\left(\frac{\mathrm{cos}\:{x}}{\mathrm{2}}+\frac{\sqrt{\mathrm{3}}\:\mathrm{sin}\:{x}}{\mathrm{2}}\right)\left(\frac{\mathrm{cos}\:{x}}{\mathrm{2}}−\frac{\sqrt{\mathrm{3}}\:\mathrm{sin}\:{x}}{\mathrm{2}}\right) \\ $$$$\mathrm{cos}\:\mathrm{3}{x}\:=\:\mathrm{cos}\:{x}\:\left(\mathrm{cos}^{\mathrm{2}} \:{x}−\mathrm{3}\:\mathrm{sin}^{\mathrm{2}} \:{x}\right) \\ $$$$\mathrm{4}\:\mathrm{cos}^{\mathrm{2}} \:{x}−\mathrm{3}\:=\:\mathrm{cos}^{\mathrm{2}} \:{x}−\mathrm{3}\:\mathrm{sin}^{\mathrm{2}} \:{x} \\ $$$$\mathrm{3}−\mathrm{3}\:=\mathrm{0} \\ $$$$\mathrm{0}=\mathrm{0}\:\checkmark \\ $$$$\Rightarrow{any}\:{x}\:{fulfills}\:{the}\:{equation}\:{and}\:{is} \\ $$$${a}\:{solution}. \\ $$

Leave a Reply

Your email address will not be published.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