Q190 / 270 · NDA 2012 (Phase 1) · Mathematics

If $\tan ^{-1} 2, \tan ^{-1} 3$ are two angles of a triangle, then what is the third angle?

  • A
    $\tan ^{-1} 2$
  • B
    $\tan ^{-1} 4$
  • Correct answer
    C
    $\pi / 4$
  • D
    $\pi / 3$

Solution

$\begin{gathered}\theta_1=\tan ^{-1} 2 \\ \theta_2=\tan ^{-1} 3, \theta_3=? \\ \theta_1+\theta_2+\theta_3=180^{\circ} \\ \tan ^{-1} 2+\tan ^{-1} 3+\theta_3=180^{\circ} \\ \tan ^{-1}\left(\frac{2+3}{1-2.3}\right)+\theta_3=180^{\circ} \\ \tan ^{-1}\left(\frac{5}{-5}\right)+\theta_3=180^{\circ} \\ 135+\theta_3=180^{\circ} \\ \theta_3=45 .\end{gathered}$

Hence, the correct answer is option: 'C'.

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