Question

solve each equation on the interval

$0\le \theta <2\pi$

$\sin \theta +\cos \theta =\sqrt{2}$

Short answer

$\theta =2n\pi +\frac{\pi }{4}$

Step-by-step solution

Step 1. The information given is

a trignometric equation

$\sin \theta +\cos \theta =\sqrt{2}$

Step 2.Multiplying equation by 12

$$\begin{gathered} \sin \theta \sin \left(\frac{\pi }{4}\right)+\cos \theta \cos \left(\frac{\pi }{4}\right)=1 \\ \cos \left(\theta -\frac{\pi }{4}\right)=\cos \left(0\right) \end{gathered}$$

Step 3. The general equation we get is

$\theta =2n\pi +\frac{\pi }{4}$