I'm currently studying for my upcoming precalculus midterm, but I'm having troubles rationalizing denominators and simplifying--and I'm not sure if I'm approaching this correctly. If anyone could lend me pointers to simplify please let me know, I'm really weak in this area.

Use a sum or difference formula to find the exact value of the given expression:

$\displaystyle \sin \frac{7\pi}{12}$

I figured...

$\displaystyle \frac{\pi}{4} + \frac{\pi}{3}=\frac{7\pi}{12}$

Thus according to the values...

$\displaystyle \frac{\sqrt2}{2}\times\frac{\sqrt2}{2}+\frac{\sqrt 3}{2}\times\frac{1}{2}$

I totally got the wrong answer from here. How should I go about simplifying this?