InterviewSolution
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InterviewSolution
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Q. The value of is equal `sum_(k=1)^13(1/(sin(pi/4+(k-1)pi/6)sin(pi/4+k pi/6))` is equalA. (a) `3-sqrt(3)`B. (b)`2(3-sqrt(3))`C. (c) `2(sqrt(3)-1)`D. (d) `2(2+sqrt3)` |
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Answer» Correct Answer - (c) Here, `sum_(k=1)^(13)1/(sin{pi/4+((k-1)pi)/6}sin (pi/4+(kpi)/6))` Converting into differences, by multiplying and dividing by `sin [(pi/4+(kpi)/6)-{pi/4+((k-1)pi)/6}],i.e.sin (pi/6).` `therefore sum_(k=1)^(13)(sin[(pi/4=kpi/6)-{pi/4+(k-1)pi/6}])/(sin frac {pi}{6} {sin{pi/4+(k-1)pi/6}sin (pi/4+kpi/6)})` `=2sum_(k=1)^13 ([sin(pi/4+(kpi)/6)cos{pi/4+(k-1)pi/6} -sin{pi/4+(k-1)pi/6}cos(pi/4+(kpi)/6)])/(sin{pi/4+(k-1)pi/6}-cot (pi/4+kpi/6))` `=2sum_(k=1)^13 [cot{pi/4+(k-1)pi/6}-cot(pi/4+k pi/6)]` `=2 [{cot(pi/4)-cot(pi/4+ pi/6)}` `+ {cot(pi/4+pi/6)-cot(pi/4+ (2pi)/6)}` `+...+ {cot(pi/4+12 pi/6)-cot(pi/4+ 13pi/6)}]` `= 2 {cot frac{pi}{4}-cot(pi/4+ 13pi/6)}` `= 2 [1-cot ((29pi)/12)]=2 [1-cot(2pi+(5pi)/12)]` `=2[1-cot frac{5pi}{12}] [therefore cot frac {5pi}{12}=(2-sqrt3)]` `=2(1-2+sqrt(3))` `=2(sqrt(3)-1)` |
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