Answer:
[tex]\displaystyle \frac{\sqrt{2} + \sqrt{6}}{4}[/tex]
Explanation:
If you recall the unit circle [from the polar graph], you would have no trouble at all figuring this out, but sinse you have trouble, do not worry about it. So, here is what you should have realised:
[tex]\displaystyle \boxed{\frac{\pi}{2}} = \frac{6}{12}\pi[/tex]
If you did not notise, [tex]\displaystyle \frac{7}{12}\pi[/tex]is just [tex]\displaystyle \frac{\pi}{12}[/tex]more than [tex]\displaystyle \frac{\pi}{2},[/tex]which means the exact value could possibly have a 4 in its denominatour [which obviously also has a desimal value] or could be irrational [values that cannot be written as fractions]. In this case, according to the unit circle, you will have a fraction, and that will be this:
[tex]\displaystyle \frac{\sqrt{2} + \sqrt{6}}{4}[/tex]
It is all about memorisation of the unit circle, which I know is difficult, but you will get used to it soon.
*Now, if you had to find [tex]\displaystyle cos\:\frac{7}{12}\pi,[/tex]then the exact value would be the OPPOCITE of the exact value of [tex]\displaystyle sin\:\frac{7}{12}\pi,[/tex]which is [tex]\displaystyle -\frac{\sqrt{2} + \sqrt{6}}{4},[/tex]because you would be crossing into the second quadrant where the x-coordinates are negative, accourding to both the cartesian and polar graphs.
I am joyous to assist you at any time.
