InterviewSolution
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InterviewSolution
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Find `x`satisfying `[tan^(-1)x]+[cos^(-1)x]=2,`where `[]`represents the greatest integer function. |
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Answer» `0 lt cot^(-1) x lt pi and -pi//2 lt tan^(-1) x lt pi//2` `rArr [cot^(-1) x] in {0, 1, 2, 3} and [tan^(-1) x] in {-2, -1, 0, 1}` For `[tan^(-1) x] + cot^(-1) x] = 2`, following cases are possible Cose(i): `[cot^(-1)x] = tan^(-1) x] = 1` `rArr 1 le cot^(-1) x lt 2 and 1 le tan^(-1) x lt pi//2` `rArr x in (cot 2, cot 1] and x in [tan 1, oo)` `:. x in phi " as " cot 1 lt tan 1` Case (ii): `[cot^(-1) x] = 2, [tan^(-1) x] = 0` `rArr 2 le cot^(-1) x lt 3 and 0 le tan^(-1) x lt 1` `rArr x in (cot 3, cot 2] and x in [0, tan 1)` `:. x in phi " as " cot 2 lt 0` So no solution Case (iii) : `[cot^(-1) x] = 3, [tan^(-1) x] = -1` `rArr 3 le cot^(-1) x lt pi and - 1 le tan^(-1) x lt 0` `rArr x in (-oo, cot 3] and x in [- tan 1, 0)` `:. x in phi cot 3 lt - tan 1` Therefore, no such value of x exist |
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