Find an acute angle $\gamma$ such that $\sin \gamma + \cos \gamma= \sqrt{2}$ 
Find an acute angle $\gamma$ such that 
  $$
\sin \gamma + \cos \gamma = \sqrt{2}
$$

 A: $$
\frac{1}{\sqrt{2}}\sin x+\frac{1}{\sqrt{2}}\cos x=1
$$
$$
\cos \frac{\pi}{4}\cos x+\sin \frac{\pi}{4}\sin x=1
$$
$$
\cos (x-\frac{\pi}{4})=1
$$
You can do it from here?
A: Hint:
$\sin \gamma + \cos \gamma $
$= \sqrt2\left(\frac{1}{\sqrt 2}\sin \gamma + \frac{1}{\sqrt 2} \cos \gamma\right) $
$= \sqrt 2 \sin \left(\gamma + \frac{\pi}{4}\right)$
A: Here is a trick that can be generalized to simplify the sum of a multiple of the sine of an angle with a multiple of the cosine of the same angle.
$$\sin x + \cos x = \sqrt 2$$
$$\sin x\cdot\frac{\sqrt 2}{2} + \frac{\sqrt 2}{2}\cdot\cos x= 1$$
$$\sin x\cdot\cos 45° + \sin 45°\cdot\cos x= 1$$
$$\sin(x+45°)=1$$
You should be able to finish from here.
A: ${1 \over \sqrt 2} \left( \sin\gamma + \cos\gamma\right) = 1$
or equivalently $$\sin\gamma\cos(\pi/4) + \cos\gamma\sin(\pi/4) = 1$$ Thus  $\sin(\gamma + \pi/4) = 1$. 
Hence $$\gamma + \pi/4 = \pi/2 + 2k\pi, \ k \text{  integer}$$ So if $\gamma$ is acute, the only solution is
$$\gamma = \ ...$$
A: Given that $$\sin \gamma+\cos\gamma=\sqrt{2}$$ $$\implies \frac{1}{\sqrt{2}}\sin \gamma +\frac{1}{\sqrt{2}}\cos\gamma=1 $$ $$\implies \sin \gamma \cos \frac{\pi}{4}+\cos\gamma \sin \frac{\pi}{4}=1 $$ $$\implies  \sin \left(\gamma+\frac{\pi}{4}\right)=1 =\sin \frac{\pi}{2}$$ writing the general solution of the above equation, we get  $$\gamma+\frac{\pi}{4}=2n\pi+\frac{\pi}{2}$$ $$\implies \gamma=2n\pi+\frac{\pi}{4}$$Where $n$ is any integer. As the unknown angle $\gamma$ is acute ($\color{red}{0<\gamma<\frac{\pi}{2}}$) hence  $$\gamma=2(0)\pi+\frac{\pi}{4}$$$$\implies \color {blue}{\gamma=\frac{\pi}{4}}$$
A: we can write 
$$\sqrt 2 = \sin t + \cos t = \sqrt 2\left(\cos \left(\frac{\pi}4\right) \cos t + \sin \left(\frac{\pi}4\right) \sin t \right) =  \sqrt 2 \cos \left(t - \pi/4\right)$$
so that we have $$\cos(t - \pi/4) = 1 \implies t - \pi/4 =   2k\pi   $$
that is $$t = 2k\pi + \frac{\pi}4, \text{ where $k$ is any integer.}$$
