Menu Close

how-do-you-simply-sin-tan-1-3x-cos-1-x-




Question Number 80113 by jagoll last updated on 31/Jan/20
how do you simply  sin (tan^(−1) (3x)+cos^(−1) (x)) ?
$${how}\:{do}\:{you}\:{simply} \\ $$$$\mathrm{sin}\:\left(\mathrm{tan}^{−\mathrm{1}} \left(\mathrm{3}{x}\right)+\mathrm{cos}^{−\mathrm{1}} \left({x}\right)\right)\:? \\ $$
Commented by john santu last updated on 31/Jan/20
let x= cos t , 3x=tan y  ⇒sin ycos t+cos ysin t   = ((3x^2 )/( (√(1+9x^2 )))) + ((√(1−x^2 ))/( (√(1+9x^2 ))))  =((3x^2 +(√(1−x^2 )))/( (√(1+9x^2 ))))
$${let}\:{x}=\:\mathrm{cos}\:{t}\:,\:\mathrm{3}{x}=\mathrm{tan}\:{y} \\ $$$$\Rightarrow\mathrm{sin}\:{y}\mathrm{cos}\:{t}+\mathrm{cos}\:{y}\mathrm{sin}\:{t}\: \\ $$$$=\:\frac{\mathrm{3}{x}^{\mathrm{2}} }{\:\sqrt{\mathrm{1}+\mathrm{9}{x}^{\mathrm{2}} }}\:+\:\frac{\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}{\:\sqrt{\mathrm{1}+\mathrm{9}{x}^{\mathrm{2}} }} \\ $$$$=\frac{\mathrm{3}{x}^{\mathrm{2}} +\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}{\:\sqrt{\mathrm{1}+\mathrm{9}{x}^{\mathrm{2}} }}\: \\ $$

Leave a Reply

Your email address will not be published.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