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Question
The trigonometric equation tan–1x = 3tan–1 a has solution for ______.
Options
`|a| ≤ 1/sqrt(3)`
`|a| > 1/sqrt(3)`
`|a| < 1/sqrt(3)`
all real value of a.
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Solution
The trigonometric equation tan–1x = 3tan–1a has solution for `underlinebb(|a| < 1/sqrt(3))`.
Explanation:
To solve the equation tan–1(x) = 3tan–1(a), we use the tangent function properties and transformations.
Let θ = tan–1(a).
Then:
x = tan(3θ)
Using the triple-angle formula for tangent:
tan(3θ) = `(3tan(θ) - tan^3(θ))/(1 - 3tan^3(θ))`
Since tan(θ) = a, substituting a in gives us:
x = `(3a - a^3)/(1 - 3a^2)`
For the function tan−1(x) = 3tan−1(a) to have a solution, the argument of tan (which is 3θ) must be within the range of the tan function, which is `(-π/2, π/2)`.
Therefore, 3θ must also be `(-π/2, π/2)`.
Given that θ = tan−1(a) is within `(-π/2, π/2)`, the condition for 3θ to remain in this interval is:
`-π/6 < θ < π/6`
This translates to:
`-π/6 < tan^-1(a) < π/6`
Taking the tangent of the bounds:
`-1/sqrt(3) < a < 1/sqrt(3)`
Thus, the condition for a is:
`|a| < 1/sqrt(3)`
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