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Satement-1: if `1/2lexle1`then `cos^(-1)x-sin^(-1){x/2+sqrt(3-3x^(2))/(2)}` is equal to `(pi)/(5)` Statement-2: `sin^(-1)(2xsqrt(1-x^(2))=2sin^(-1)x if x in -(1)sqrt(2),(1)sqrt(2))`A. Statement-1 is is True, Statement-2 is true, Statement-2 is a correct explanation for Statement-1.B. Statement-1 is True, Statement-2 is True, Statement-2 is not a correct explanation for Statement-1.C. Statement-1 is True, Statement-2 is False.D. Statement-1 is False, Statement-2 is True. |
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Answer» Let x = cos `theta` then `1/2 le x le 1 rarr 1/2 le cos theta le 1 rarr 0 le theta le (pi)/(3)` `therefore -theta-sin^(-1){1/2cos theta +sqrt(2)/(2)sin theta}` `=theta -sin^(-1){sin(pi)/(6)cos theta +cos (pi)/(6) sin theta}` `=theta -sin^(-1){sin(theta+(pi)/(6))}` `=theta -(theta +(pi)/(6))=-(pi)/(6)` so statement 1 true let x =sinx `theta` then `(1)/sqrt(2) le x le (1)sqrt(2) rarr -(1)/sqrt(2) le sin theta le (1)/sqrt(2) rarr =(pi)/(4) le theta le (pi)/(4)` `there sin^(-1)2xsqrt(1-x^(2))=sin^(-1)(sin 2 theta) =2 sin^(-1)x` so statement 2 true we have `1/2 le x le 1 rarr x^(2)+3/4gt 1` using : `sin^(-1) x + sin^(-1)y =pi -sin^(-1)xsqrt(1-y^(2))+ysqrt(1-x^(2))` when `0 lt x,y le 1 and x^(2) a+ y^(2) gt1` we have `therefore sin^(-1)((x)/(2)+(sqrt(3-3x^(2))/(2))=pi-sin^(-1)x-(pi)/(3)=2pi)/(3)-=sin^(-1)x` `=cot^(-1)x=(2x)/(3)+sin^(-1)x=(pi)/(2)-(2pi)/(3)=-(pi)/(6)` so statement -1 is true statement -2 true (see theory ) |
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