Respuesta :

Answer:

+120/169 or -120/169

Step-by-step explanation:

  • let [tex]cos^{-1}[\frac{5}{13}  ] = \alpha[/tex]

where, alpha is some angle that satisfies the assumed condition.

  • so, [tex]cos\alpha= 5/13[/tex]

[ taking cos to the other side  or applying cos on both sides]

  • now, substitute this in the given expression
  • [tex]sin(2cos^-1(5/13)) = sin(2\alpha )[/tex]

as sin[tex]\beta[/tex] =  [tex]+ or -\sqrt{1-(cos\beta) ^{2}[/tex]

[by general trigonometry formula: [tex]cos^{2}[\beta]  +sin^{2}[\beta]=1[/tex]]

so if  [tex]cos\alpha= 5/13[/tex], we can get sin[tex]\alpha[/tex] from the above formula as + or - 12/13

(because, after taking square root on both sides we keep + or -]

  • as, sin[tex]2\beta  = 2*sin[\beta ]*cos[\beta ][/tex]

[by general trigonometry formula]

  • here, now [tex]sin[2\alpha ]=2*(+or- 12/13)*5/13\\[/tex]

so, the final value can be 120/169 or -120/169.

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