Question

If $\tan \theta =\frac{1}{2}$ and $\tan \varphi =\frac{1}{3}$ then find the value of $\theta +\varphi$.all angles are acute

Answer 1

Given, $\tan \theta =\frac{1}{2}$......(1) and $\tan \varphi =\frac{1}{3}$......(2).

Now,

$\tan (\theta +\varphi )$

$=\frac{\tan \theta +\tan \varphi }{1-\tan \theta .\tan \varphi }$

$=\frac{\frac{1}{2}+\frac{1}{3}}{1-\frac{1}{2}.\frac{1}{3}}$ [ Using (1) and (2)]

$=\frac{\frac{5}{6}}{\frac{5}{6}}$ $=1$.

So, $\tan (\theta +\varphi )=1$

or, $\theta +\varphi ={45}^o$. [ Taking $\theta ,\varphi$ to be acute]