Answer :
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.