Question

If `cos^-1(2/(3x)) + cos^-1(3/(4x)) = π/2(x > 3/4)`, then x is equal to ______.

पर्याय

• `sqrt(145)/12`

• `sqrt(145)/10`

• `sqrt(146)/12`

• `sqrt(145)/11`

Answer 1

If `cos^-1(2/(3x)) + cos^-1(3/(4x)) = π/2(x > 3/4)`, then x is equal to `underlinebb(sqrt(145)/12)`.

Explanation:

`cos^-1(2/(3x)) + cos^-1(3/(4x)) = π/2;(x > 3/4)`

`\implies cos^-1(2/(3x)) = π/2 - cos^-1(3/(4x))`

`\implies cos^-1(2/(3x)) = sin^-1(3/(4x))` ...`[∵ sin^-1x + cos^-1x = π/2]`

Put `sin^-1(3/(4x))` = θ

`\implies` sin θ = `3/(4x)`

`\implies` cos θ = `sqrt(1 - sin^2θ) = sqrt(1 - 9/(16x^2))`

`\implies` θ = `cos^-1(sqrt(16x^2 - 9)/(4x))`

∴ `cos^-1(2/(3x)) = cos^-1(sqrt(16x^2 - 9)/(4x))`

`\implies 2/(3x) = sqrt(16x^2 - 9)/(4x)` 

`\implies` x² = `(64 + 81)/(9 xx 16)`

`\implies` x = `±sqrt(145/144)`

`\implies` x = `sqrt(145)/12`  ...`(∵ x > 3/4)`