Question

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

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

Answer 1

Answer: (B)

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