Question

The general solution of the equation $\sin\: x + \cos\: x =1$ is given by

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

Answer 1

Answer: (C)

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