If tan (3x) = 5 tan (x), what is the value of sin (x)?

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Jhun Vert
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If tan (3x) = 5 tan (x), what is the value of sin (x)?

If tan (3x) = 5 tan (x), what is the value of sin (x)?

esmilitar
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ans. sin (x)=√2/4
solns:
tan(3x)=5tan(x) -----> eq. 1
tan(3x)=tan (2x + x)= (tan (2x)+tan(x))/(1-tan(2x)tan(x))-----> eq. 2
subs. 2 in 1
(tan (2x)+tan(x))/(1-tan(2x)tan(x))=5tan(x)
tan(2x) + tan (x)=5tan(x)-5tan(2x)tan2(x)
tan(2x)=4tan(x)-5tan(2x)tan2(x)
tan(2x)+5tan(2x)tan2(x)=4tan(x)
tan(2x)(1 + 5tan2(x)=4tan(x)----- eq. 3

tan(2x)=2tan(x)/(1-tan2(x)) -----eq.4
substitute 4 to 3
2tan(x) (1 + 5tan2(x))/(1-tan2(x)) =4tan(x)
2tan(x) (1 + 5tan2(x))=4tan(x)(1-tan2(x))
2tan(x) + 10tan3(x))=4tan(x)-4tan3(x)
14tan3(x)=2tan(x)
14(sin3(x)/cos3(x))=2(sin(x)/cos(x))
14(sin2(x)/cos2(x))=2
14sin2(x)=2cos2(x)
14sin2(x)=2(1-sin2(x))
14sin2(x)+2sin2(x)=2
16sin2(x)=2
sin(x)=√(2/16) or
sin (x)=√2/4 ans.
where x=20.705o

8eightI'sD
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