Question

Write the following function in the simplest form:

`tan^(-1) ((3a^2 x - x^3)/(a^3 - 3ax^2)), a > 0; (-a)/sqrt(3) < x < a/sqrt(3)`

Answer 1

`tan^(-1) ((3a^2x - x^3)/(a^3 - 3ax^2))`

Put x = a tan θ

⇒ `x/a = tan θ`

⇒ `θ = tan^(-1)  x/a`

`tan^(-1) ((3a^2 x - x^3)/(a^3 - 3ax^2)) `

= `tan^(-1) ((3a^2 · a tan θ - a^3 tan^3 θ)/(a^3 - 3a · a^2 tan^2 θ))`

= `tan^(-1) ((3a^3 tan θ - a^3  tan^3 θ)/(a^3 - 3a^3 tan^2 θ))`

= `tan^(-1) ((3tan θ - tan^3 θ)/(1 - 3tan^2 θ))`

= tan^–1 (tan 3θ)

= 3θ

= `3 tan^(-1)  x/a`