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INVERSE TRIGNOMETRIC FUNCTIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsINVERSE TRIGNOMETRIC FUNCTIONS3 Marks
Question
Write the following function in the simplest form:
$\tan^{-1}\bigg(\frac{3a^{2}x-x^{3}}{a^{3}-3ax^{2}}\bigg), a>0; \frac{-a}{\sqrt{3}}\leq x\leq\frac{a}{\sqrt{3}}$
✓

Answer

$\tan^{-1}\Bigg(\frac{3a^2x-x^3}{a^3-3ax^2}\Bigg)=\tan^{-1}\Bigg(\frac{3\big(\frac{x}{a}\big)-\big(\frac{x}{a}\big)^3}{1-3\big(\frac{x}{a}\big)^2}\Bigg)$ [Dividing numerator and denominator by $a^3$]
Putting $\frac{x}{a}=\tan\theta$ so that $\theta=\tan^{-1}\frac{x}{a}$
$=\tan^{-1}\bigg(\frac{3\tan\theta-\tan^3\theta}{1-3\tan^2\theta}\bigg)=\tan^{-1}\tan3\theta$
$=3\theta=3\tan^{-1}\frac{x}{a}$

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