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If sin^2 theta=1/4 , then the value of t...

If `sin^2 theta=1/4` , then the value of `theta` is

A

`npi +- pi/6`

B

`npi+-pi/4`

C

`npi + pi/3`

D

`npi`

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A
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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))/(cot^(2)""(pi)/(7) + cot^(2)""(2pi)/(7) + cot^(2)""(3pi)/(7)) is

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

AAKASH INSTITUTE ENGLISH-TRIGNOMETRIC FUNCTIONS -Assignment section-A (Objective Type Questions (One option is correct))
  1. If sintheta +costheta =1, then the value of sin2theta is equal to

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  2. If "tan"A=1/(2),"tan"B=1/(3), then "tan"(2A+B) is equal to

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  3. The minute hand of a watch is 3 cm long. How far does its tip move in ...

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  4. If sintheta=(-4)/(5) and theta lies in third quadrant, then the value...

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  5. The value of tan75^(@) =

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  6. If omega is a non-real cube root of unity and n is not a multiple o...

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  7. The value of 2sinA cos^(3)A -2sin^(3)A cosA is

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  8. If tanalpha=(sqrt(3)+1)/(sqrt(3)-1), then the expression cos 2alpha +(...

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  9. If sin^2 theta=1/4 , then the value of theta is

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  10. The solution of the equation cos^2theta+sintheta+1=0 lies in the inter...

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  11. The general solution of tan3x=1 is

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  12. If sin theta =sqrt(3) cos theta, -pi lt theta lt 0, then the value of ...

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  13. Number of solution of the equation tanx+secx=2cosx lying in the interv...

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  14. If the lengths of the sides of a triangle are 3, 5, 7, then its larges...

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  15. The value of sin^2 75^0-sin^2 15^0 is a. 1//2 b. sqrt(3)//2 c. 1 d. 0

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  16. If a cosA=b cosB, then triangleABC is

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  17. If a=16, b=24 and c=20, then the value of cos(B/2) is

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  18. If a=1+sqrt(3), b=2 and c=sqrt(2), then the measure of angleB is

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  19. If angleA=60^(@), b=2 and c=4, then the value of a is

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  20. If b=sqrt(3), c=1 and angleA=30^(@), then the measure of angleB is

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