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If $\tan ^{-1} 2, \tan ^{-1} 3$ are two angles of a triangle, then what is the third angle?
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Verified Answer
The correct answer is:
$\pi / 4$
Let $\mathrm{A}=\tan ^{-1} 2, \mathrm{~B}=\tan ^{-1} 3$ and $\mathrm{C}$ be the angles of a
triangle.
By angle sum property, we have
$\tan ^{-1} 2+\tan ^{-1} 3+\mathrm{C}=180^{\circ}$
$\Rightarrow \tan ^{-1}\left(\frac{5}{-5}\right)=180^{\circ}-\mathrm{C}$
$\Rightarrow \tan ^{-1}(-1)=180^{\circ}-\mathrm{C}$
$\Rightarrow \frac{3 \pi}{4}=\pi-\mathrm{C} \Rightarrow \mathrm{C}=\pi-\frac{3 \pi}{4}=\frac{\pi}{4}$
Hence, third angle is $\frac{\pi}{4}$.
triangle.
By angle sum property, we have
$\tan ^{-1} 2+\tan ^{-1} 3+\mathrm{C}=180^{\circ}$
$\Rightarrow \tan ^{-1}\left(\frac{5}{-5}\right)=180^{\circ}-\mathrm{C}$
$\Rightarrow \tan ^{-1}(-1)=180^{\circ}-\mathrm{C}$
$\Rightarrow \frac{3 \pi}{4}=\pi-\mathrm{C} \Rightarrow \mathrm{C}=\pi-\frac{3 \pi}{4}=\frac{\pi}{4}$
Hence, third angle is $\frac{\pi}{4}$.
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