If + =1, then find the general value of - Vidyadip

Question

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

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Answer

$\sin\theta+\cos\theta=1$
$\Rightarrow\frac{1}{\sqrt2}\sin\theta+\frac{1}{\sqrt2}\cos\theta=\frac{1}{\sqrt2}\Rightarrow\sin\theta\cos\frac{\pi}{4}+\cos\theta\sin\frac{\pi}{4}=\frac{1}{\sqrt2}$
$\Rightarrow\sin\Big(\theta+\frac{\pi}{4}\Big)=\sin\frac{\pi}{4}\Rightarrow\theta+\frac{\pi}{4}=\text{n}\pi+(-1)^\text{n}\frac{\pi}{4},\text{n}\in\text{Z}$
$\therefore\theta=\text{n}\pi(-1)^\text{n}\frac{\pi}{4}-\frac{\pi}{4},\text{n}\in\text{Z}$

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