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Find the value of`sin^(-1)(2^x)`(ii) `cos^(-1)sqrt(x^2-x+1)``tan^(-1)(x^2)/(1+x^2)`(iv) `sec^(-1)(x+1/x)` |
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Answer» (i) We have that `2^(@) gt 0` But for `sin^(-1) (2^(x))` to be defined, we must have `2^(x) le 1` `:. 0 lt 2^(x) le 1` `rArr 0 sin^(-1) (2^(x)) le pi//2` (ii) `cot^(-1) sqrt(x^(2)-x+1)= cos^(-1) sqrt((x-1/2)^(2)+3/4)` We have `sqrt((x-1/2)^(2)+3/4) ge sqrt(3)/2` `:. sqrt(3)/2 le sqrt((x-1/2)^(2)+3/4) le 1` `:. 0 le cos^(-1) sqrt((x-1/2)^(2)+3/4) le pi/6` (iii) `"tan"^(-1) x^(2)/(1+x^(2))=tan^(-1) (1-1/(1+x^(2)))` Now, `1 le 1+ x^(2) lt oo` `:. 0 lt 1/(1+x^(2)) lt 1` `:. -1 lt -1/(1+x^(2)) lt 0` `:. 0 lt 1-1/(1+x^(2)) lt 1` `:. 0 lt tan^(-1) (1-1/(1+x^(2))) lt pi/4` (iv) Let `x+1/x=y` `:. x^(2)-yx+1=0` Since x is real, `D ge 0` `:. y^(2) -4 ge 0` `:. y le -2` or `y ge 2` `:. sec^(-1) (x+1/x) in [pi/3, pi/2) uu (pi/2, (2pi)/3]` |
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