find trigo ratio
Since $\displaystyle \frac{\sqrt{3}+1}{2\sqrt{2}}=\frac{1}{2}\cdot\frac {\sqrt{2}}{2}+\frac{\sqrt{3}}{2}\cdot\frac{\sqrt{2 }}{2}$,
we can break it up and use ones we do know by using the addition formula
$\displaystyle sin(\frac{\pi}{6})=\frac{1}{2}$
$\displaystyle cos(\frac{\pi}{4})=\frac{\sqrt{2}}{2}$
$\displaystyle sin(\frac{\pi}{4})=\frac{\sqrt{2}}{2}$
$\displaystyle cos(\frac{\pi}{6})=\frac{\sqrt{3}}{2}$
Now, use the addition formula to put it together
It would appear that when we add them up we get
$\displaystyle \frac{\pi}{6}+\frac{\pi}{4}=\frac{5\pi}{12}$
So, $\displaystyle sin(\frac{5\pi}{12})$
See how to do that now?. Sometimes they're not so obvious so we have to do some gymnastics.
addition formula for sine: $\displaystyle sin(a+b)=sin(a)cos(b)+cos(a)sin(b)$
$\displaystyle sin(\frac{\pi}{6}+\frac{\pi}{4})=sin(\frac{\pi}{6} )cos(\frac{\pi}{4})+cos(\frac{\pi}{6})sin(\frac{\p i}{4})=sin(\frac{5\pi}{12})$
Put them together from the last post. and you get
$\displaystyle \frac{\sqrt{3}+1}{2\sqrt{2}}\approx{0.965925....}$
or 75 degrees $\displaystyle sin(75)=\frac{\sqrt{3}+1}{2\sqrt{2}}$