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The value of sin""(2pi)/(7)+sin""(4pi)/(...

The value of `sin""(2pi)/(7)+sin""(4pi)/(7)+sin""(8pi)/(7),` is

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The correct Answer is:
`(sqrt(7))/(2)`
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To find the sum sin^(2) ""(2pi)/(7) + sin^(2)""(4pi)/(7) +sin^(2)""(8pi)/(7) , we follow the following method. Put 7theta = 2npi , where n is any integer. Then " " sin 4 theta = sin( 2npi - 3theta) = - sin 3theta This means that sin theta takes the values 0, pm sin (2pi//7), pmsin(2pi//7), pm sin(4pi//7), and pm sin (8pi//7) . From Eq. (i), we now get " " 2 sin 2 theta cos 2theta = 4 sin^(3) theta - 3 sin theta or 4 sin theta cos theta (1-2 sin^(2) theta)= sin theta ( 4sin ^(2) theta -3) Rejecting the value sin theta =0 , we get " " 4 cos theta (1-2 sin^(2) theta ) = 4 sin ^(2) theta - 3 or 16 cos^(2) theta (1-2 sin^(2) theta)^(2) = ( 4sin ^(2) theta -3)^(2) or 16(1-sin^(2) theta) (1-4 sin^(2) theta + 4 sin ^(4) theta) " " = 16 sin ^(4) theta - 24 sin ^(2) theta +9 or " " 64 sin^(6) theta - 112 sin^(4) theta - 56 sin^(2) theta -7 =0 This is cubic in sin^(2) theta with the roots sin^(2)( 2pi//7), sin^(2) (4pi//7), and sin^(2)(8pi//7) . The sum of these roots is " " sin^(2)""(2pi)/(7) + sin^(2)""(4pi)/(7) + sin ^(2)""(8pi)/(7) = (112)/(64) = (7)/(4) . The value of (tan^(2)""(pi)/(7) + tan^(2)""(2pi)/(7) + tan^(2)""(3pi)/(7))xx (cot^(2)""(pi)/(7) + cot^(2)""(2pi)/(7) + cot^(2)""(3pi)/(7)) is

ARIHANT MATHS-TRIGONOMETRIC FUNCTIONS AND IDENTITIES-Exercise (Questions Asked In Previous 13 Years Exam)
  1. The value of sin""(2pi)/(7)+sin""(4pi)/(7)+sin""(8pi)/(7), is

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  2. If alpha and beta are non-zero real number such that 2(cos beta-cos al...

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  3. Let -1/6 < theta < -pi/12 Suppose alpha1 and beta1, are the root...

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  4. The value of sum(k=1)^(13) (1)/(sin(pi/4 + ((k-1)pi)/(6))sin(pi/4 + (k...

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  5. Let f : (-1, 1) -> R be such that f(cos4theta) = 2/(2-sec^2theta for t...

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  6. The number of all possible values of theta, where 0 lt theta lt pi, f...

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  7. For 0 lt theta lt pi/2, the solution (s) of sum(m=1)^(6) cosec (the...

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  8. If sin^ 4 x/2+cos^4 x/3 =1/5 then

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  9. Let theta in (0,pi/4) and t1=(tan theta)^(tan theta), t2=(tan theta)6(...

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  10. cos(alpha-beta)=1a n dcos(alpha+beta)=l/e , where alpha,betamu in [-pi...

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  11. If 5(tan^2x - cos^2x)=2cos 2x + 9, then the value of cos4x is

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  12. Let fk(x) = 1/k(sin^k x + cos^k x) where x in RR and k gt= 1. Then f4(...

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  13. The expression (tanA)/(1-cotA)+(cotA)/(1-tanA) can be written as (1) s...

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  14. If a Delta PQR " if" 3 sin P + 4 cos Q = 6 and 4 sin Q + 3 cos P =1 , ...

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  15. If A = sin^2x + cos^4 x, then for all real x :

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  16. Let cos (alpha+beta) = 4/5 and sin(alpha-beta)=5/13 where 0<= alpha,...

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  17. If cosalpha+cosbeta+cosgamma=0=sinalpha+sinbeta+singamma, then which...

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  18. A triangular park is enclosed on two sides by a fence and on the third...

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  19. If 0 lt x lt pi and cos x + sin x = 1/2, then tan x is

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  20. In Delta PQR , /R=pi/4, tan(P/3), tan(Q/3) are the roots of the equati...

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