Question

The general solution of $\sin x+\cos x=1$ is

• A. $x=2n\pi ,n\in Z$
• B. $x=2n\pi +\frac{\pi }{2},n\in Z$
• C. $x=\left\{2n\pi \right\}\cup \{2n\pi +\frac{\pi }{2}\},n\in Z$
• D. $x=\varphi$

Answer 1

The correct option is C $x=\left\{2n\pi \right\}\cup \{2n\pi +\frac{\pi }{2}\},n\in Z$

$\sin x+\cos x=1\Rightarrow \frac{1}{\sqrt{2}}\sin x+\frac{1}{\sqrt{2}}\cos x=\frac{1}{\sqrt{2}}\Rightarrow \sin x\sin \left(\frac{\pi }{4}\right)+\cos x\cos \left(\frac{\pi }{4}\right)=\frac{1}{\sqrt{2}}\Rightarrow \cos (x-\frac{\pi }{4})=\frac{1}{\sqrt{2}}\Rightarrow x-\frac{\pi }{4}=2n\pi \pm \frac{\pi }{4}\therefore x=\left\{2n\pi \right\}\cup \{2n\pi +\frac{\pi }{2}\},n\in Z$