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If sin(picostheta)=cos(pi sintheta),t h ...

If `sin(picostheta)=cos(pi sintheta),t h e n sin2theta=`

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To solve the equation \( \sin(\pi \cos \theta) = \cos(\pi \sin \theta) \) and find the value of \( \sin 2\theta \), we can follow these steps: ### Step-by-step Solution: 1. **Rewrite the Equation**: We start with the given equation: \[ \sin(\pi \cos \theta) = \cos(\pi \sin \theta) ...
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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

RD SHARMA ENGLISH-TRIGONOMETRIC RATIOS OF COMPOUND ANGLES-All Questions
  1. Prove that: (sinx)/(cos3x)+(sin3x)/(cos9x)+(sin9x)/(cos27 x)=(1/2)(ta...

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  2. Prove that: 1/(cos(x-a)cos(x-b))=(tan(x-b)-"tan"(x-a))/(sin(a-b))

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  3. If sin(picostheta)=cos(pi sintheta),t h e n sin2theta=

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  4. If at a nalpha+bt a nbeta=(a+b)tan((alpha+beta)/2) , where alpha!=beta...

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  5. Prove that: 1/(sin(x-a)sin(x-b))=(cot(x-a)-"cot"(x-b))/(sin(a-b))

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  6. If tantheta=1/2a n dt a nvarphi=1/3, then the value of theta+varphi ...

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  7. If tantheta=(sinalpha-cosalpha)/(sinalpha+cosalpha) then show that si...

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  8. Prove that: 1/(sin(x-a)cos(x-b))=(cot(x-a)+"tan"(x-b))/(cos(a-b))

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  9. Find the maximum and minimum values of the following expressions. (i)...

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  10. If sinA=3/5 and cosB=9/41, 0 < A < pi/2 and 0 < B < pi/2 then find the...

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  11. If cos A=4/(5), cosB=12/13 and (3pi)/2 < A, B < 2pi then find the valu...

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  12. Find the values of the following: t a n 15^0 ii. t a n 75^0

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  13. Find the values of the following: (i)tan105^0 ( ii.) tan((13pi)/12)

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  14. Evaluate the following: sin(7pi)/(12)cospi/4-cos(7pi)/(12)sinpi/4

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  15. Evaluate the following: cos(2pi/3)cos(pi/4)-sin(2pi/3)sin(pi/4)

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  16. Prove that: cos(pi/4-A)cos(pi/4-B)-sin(pi/4-A)sin(pi/4-B)="sin"(A+B)

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  17. Prove that: sin(n+1)A sin(n+2)A+cos(n+1)A cos(n+2)A=cos A

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  18. Prove that: cos(pi/4+x)+cos(pi/4-x\ )=sqrt(2)cosx

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  19. Prove that: cos((3pi)/4+x)-cos((3pi)/4-x\ )=sqrt(2)sinx

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  20. Prove that: sin(x+y)/sin(x-y)=(tanx+tany)/(tanx-tany)

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