Question

If tan^-12, tan^-13 are measures of two angles of triangle, then the measure of its third angle is _________________.

Answer 1

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

Now,

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

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

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

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

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

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

If tan⁻¹2, tan⁻¹3 are measures of two angles of triangle, then the measure of its third angle is $\underline{\frac{\mathrm{\pi }}{4}}$.