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1+sinx+sin^(2)x+..."to"oo=2sqrt3 + 4, if...

`1+sinx+sin^(2)x+..."to"oo=2sqrt3 + 4, if` x = ?

A

`x=(2pi)/(3)or,(pi)/(3)`

B

`x=(7pi)/(6)`

C

`x=(pi)/(6)`

D

`x=(pi)/(4)`

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To solve the equation \(1 + \sin x + \sin^2 x + \ldots = 2\sqrt{3} + 4\), we can follow these steps: ### Step 1: Identify the series The left-hand side of the equation is an infinite series: \[ S = 1 + \sin x + \sin^2 x + \ldots \] This series is a geometric progression (GP) where the first term \( a = 1 \) and the common ratio \( r = \sin x \). ### Step 2: Write the formula for the sum of an infinite GP The sum \( S \) of an infinite geometric series can be calculated using the formula: \[ S = \frac{a}{1 - r} \] where \( |r| < 1 \). ### Step 3: Substitute the values into the formula Substituting \( a = 1 \) and \( r = \sin x \): \[ S = \frac{1}{1 - \sin x} \] ### Step 4: Set the equation equal to the given value Now we set the sum equal to the right-hand side of the original equation: \[ \frac{1}{1 - \sin x} = 2\sqrt{3} + 4 \] ### Step 5: Solve for \( \sin x \) First, we simplify the right-hand side: \[ 2\sqrt{3} + 4 = 2\sqrt{3} + 4 \] Now, cross-multiply to eliminate the fraction: \[ 1 = (2\sqrt{3} + 4)(1 - \sin x) \] Expanding this gives: \[ 1 = 2\sqrt{3} + 4 - (2\sqrt{3} + 4)\sin x \] ### Step 6: Rearrange the equation Rearranging the equation: \[ (2\sqrt{3} + 4)\sin x = 2\sqrt{3} + 4 - 1 \] \[ (2\sqrt{3} + 4)\sin x = 2\sqrt{3} + 3 \] ### Step 7: Isolate \( \sin x \) Now, divide both sides by \( 2\sqrt{3} + 4 \): \[ \sin x = \frac{2\sqrt{3} + 3}{2\sqrt{3} + 4} \] ### Step 8: Rationalize the expression To rationalize the expression, multiply the numerator and denominator by the conjugate of the denominator: \[ \sin x = \frac{(2\sqrt{3} + 3)(2\sqrt{3} - 4)}{(2\sqrt{3} + 4)(2\sqrt{3} - 4)} \] Calculating the denominator: \[ (2\sqrt{3})^2 - 4^2 = 12 - 16 = -4 \] Calculating the numerator: \[ 2\sqrt{3} \cdot 2\sqrt{3} - 8 + 6\sqrt{3} - 12 = 6\sqrt{3} - 8 \] Thus: \[ \sin x = \frac{6\sqrt{3} - 8}{-4} = -\frac{3\sqrt{3}}{2} + 2 \] ### Step 9: Find the angles corresponding to \( \sin x \) We know that \( \sin x = \frac{\sqrt{3}}{2} \) corresponds to angles \( x = \frac{\pi}{3} \) or \( x = \frac{2\pi}{3} \). ### Final Answer Thus, the values of \( x \) are: \[ x = \frac{\pi}{3} \text{ or } x = \frac{2\pi}{3} \]
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OBJECTIVE RD SHARMA ENGLISH-TRIGONOMETRIC RATIOS AND IDENTITIES-Exercise
  1. If cosA+cosB+cosC=0, then cos3A+cos3B+cos3C is equal to

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  2. The minimum value of cos 2 theta + cos theta for all real values of th...

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  3. cos9^@-sin9^@=

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  4. If y=(sin 3theta)/(sintheta), theta nen pi, then

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  5. about to only mathematics

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  6. about to only mathematics

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  7. 1+sinx+sin^(2)x+..."to"oo=2sqrt3 + 4, if x = ?

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  8. If sin(x-y)=cos(x+y)=1/2, the value of x and y lyingbetween 0^(@)and 9...

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  9. If alpha and beta be between 0 and (pi)/(2)and if cos(alpha+beta)=(12)...

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  10. If tan "" (alpha )/(2) and tan "" (beta)/( 2) are the roots of the eq...

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  11. 2x^(2) + 3x - alpha = 0 " has roots "-2 and beta " while the equation ...

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  12. If A+B+C=0, then the value of sum cot (B+C-A) cot (C+A-B) is equal to

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  13. In (0,pi//2)tan^(m)x+cot^(m)x attains

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  14. For -pi/2 lt theta lt pi/2, (sintheta + sin 2theta)/(1+costheta + cos2...

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  15. If y tan (A+B+C) =x tan (A+B-C) = gamma then tan 2C=

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  16. Which of the following is not the qyadratic equation whose roots are c...

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  17. cos alphasin(beta-gamma)+cosbetasin(gamma-alpha)+cos gammasin(alpha-be...

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  18. Prove that sin47^(@)+sin61^(@)-sin11^(@)-sin25^(@)=cos7^(@).

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  19. Write the value of cos1^0+cos2^0+cos3^0++cos 180^0dot

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  20. The value of (tan7 0^0-tan2 0^0)/(tan5 0^0)=

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