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Question
Write the following function in the simplest form:
`tan^(-1) ((3a^2 x - x^3)/(a^3 - 3ax^2)), a > 0; (-a)/sqrt3 <= x a/sqrt3`
Solution
`tan^(-1) ((3a^2x -x^3)/(a^3 - 3ax^2))`
Put `x = a tan theta`
`=> x/a = tan theta`
`=> theta`
`= tan^(-1) x/a`
`tan^(-1) ((3a^2 x - x^3)/(a^3 - 3ax^2)) `
`= tan^(-1) ((3a^2. atan theta - a^3 tan^3 theta)/(a^3 - 3a.a^2 tan^2 theta))`
`tan^(-1) ((3a^3 tan theta - a^3 tan^3 theta)/(a^3 - 3a^3 tan^2 theta))`
`tan^(-1) ((3tan theta - tan^3 theta)/(1 -3tan^2 theta))`
`= tan^(-1) (tan 3 theta)`
`= 3 theta`
`= 3 tan^(-1) x/a`
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