If tan are measures of two angles of triangle, then the measure of its third angle is

Question:

If $\tan ^{-1} 2, \tan ^{-1} 3$ are measures of two angles of triangle, then the measure of its third angle is________________.

Solution:

Let the measure of third angle of the triangle be x.

Now,

$\tan ^{-1} 2+\tan ^{-1} 3+x=\pi$                                                 (Angle sum property of triangle)

$\Rightarrow \pi+\tan ^{-1}\left(\frac{2+3}{1-2 \times 3}\right)+x=\pi$                        $\left[\tan ^{-1} x+\tan ^{-1} y=\pi+\tan \left(\frac{x+y}{1-x y}\right)\right.$, if $\left.x y>1\right]$

$\Rightarrow \tan ^{-1}(-1)+x=0$

$\Rightarrow-\frac{\pi}{4}+x=0$

$\Rightarrow x=\frac{\pi}{4}$

Thus, the measure of third angle of the triangle is $\frac{\pi}{4}$.

If $\tan ^{-1} 2, \tan ^{-1} 3$ are measures of two angles of triangle, then the measure of its third angle is $\frac{\pi}{4}$

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