Question

If $\sin \,\theta+\cos \, \theta=1$, then the general value of $\theta$ is

• A. $2 n \pi$
• B. $n \pi+(-1)^{ n } \frac{\pi}{4}-\frac{\pi}{4}$
• C. $2 n \pi+\frac{\pi}{2}$
• D. $(2 n-1)+\frac{\pi}{4}$

Answer 1

Answer: (B)

$\sin \theta+\cos \theta=1$.
Dividing by $\sqrt{1^{2}+1^{2}}=\sqrt{2}$,
$ \frac{1}{\sqrt{2}} \sin \theta+\frac{1}{\sqrt{2}} \cos \theta=\frac{1}{\sqrt{2}} $
$\Rightarrow \sin \left(\theta+\frac{\pi}{4}\right)=\frac{1}{\sqrt{2}}=\sin \frac{\pi}{4} $
$\Rightarrow \theta+\frac{\pi}{4}= n \pi+(-1)^{ n } \frac{\pi}{4} $
$\Rightarrow \theta= n \pi+(-1)^{ n } \frac{\pi}{4}-\frac{\pi}{4}$ .