$\sin \left( x+\frac{3\pi }{4} \right)=\operatorname{sinx}\cos \frac{3\pi }{4}+\operatorname{cosx}\sin \frac{3\pi }{4}$ 

$=\left( \operatorname{sinx} \right)\left( \frac{-1}{\sqrt{2}} \right)+\left( \operatorname{cosx} \right)\left( \frac{1}{\sqrt{2}} \right)=\frac{-\operatorname{sinx}}{\sqrt{2}}+\frac{\operatorname{cosx}}{\sqrt{2}}\begin{matrix}    {} & \left( 1 \right)  \\ \end{matrix}$ 

توجه کنید که:

$\cos \left( \frac{3\pi }{2}+x \right)=\cos \left( \pi +\left( \frac{\pi }{2}+x \right) \right)=-\cos \left( \frac{\pi }{2}+x \right)=\operatorname{sinx}\begin{matrix}    {} & \left( 2 \right)  \\ \end{matrix}$ 

$\sqrt{2}\sin \left( x+\frac{3\pi }{4} \right)=\cos \left( \frac{3\pi }{2}+x \right)\xrightarrow{\left( 2 \right),\left( 1 \right)}-\operatorname{sinx}+\operatorname{cosx}=\operatorname{sinx}\Rightarrow 2\operatorname{sinx}=\operatorname{cosx}\Rightarrow 2tanx=1\Rightarrow tanx=\frac{1}{2}$

همچنین:

$\sin 2x=\frac{2\tan x}{1+{{\tan }^{2}}x}\xrightarrow{\tan x=\tfrac{1}{2}}\sin 2x=\sin 2x=\frac{1}{1+\frac{1}{4}}=\frac{4}{5}$