Question

If $\sin \theta +\sin \varphi =a$ and $\cos \theta +\cos \varphi =b$, then find the value of $\tan \frac{\theta +\varphi }{2}$.

Answer 1

$\sin \theta +\sin \varphi =a$ ….. (1)

$\cos \theta +\cos \varphi =b$ …… (2)

By equation (1) and (2) to,

$\sin \theta +\sin \varphi =2\sin \frac{(\theta +\varphi )}{2}\cos \frac{(\theta -\varphi )}{2}$ …… (3)

$\cos \theta +\cos \varphi =2\cos \frac{(\theta +\varphi )}{2}\cos \frac{(\theta -\varphi )}{2}$ …… (4)

Dividing equation (3) and (4) to, we get

$\frac{\sin \theta +\sin \varphi }{\cos \theta +\cos \varphi }=\frac{2\sin \frac{(\theta +\varphi )}{2}\cos \frac{(\theta -\varphi )}{2}}{2\cos \frac{(\theta +\varphi )}{2}\cos \frac{(\theta -\varphi )}{2}}$

$\frac{a}{b}=\frac{\sin \frac{(\theta +\varphi )}{2}}{\cos \frac{(\theta +\varphi )}{2}}$

$\tan \frac{(\theta +\varphi )}{2}=\frac{a}{b}$

Hence, this is the answer.