Question 570994
It does not take a genius to find the exact value of {{{sin(7*pi/12)}}}, but having those trigonometric identity formulas handy helps.
The expression you posted for {{{sin(7*pi/12)}}} is wrong. Either someone made a typo somewhere, or the person who wrote the problem is trying to confuse us all.
Since the expression given for {{{sin(7*pi/12)}}} looked fishy to me, I went looking for the trigonometric identity formulas to find the correct exact value of {{{sin(7*pi/12)}}}.
It turns out that {{{highlight(sin(7*pi/12)=(sqrt(2)+sqrt(6))/2))}}}
HOW I CALCULATED THAT (just in case you care)
I found the trigonometric identity
{{{sin(A+B)=sin(A)cos(B)+cos(A)sin(B)}}}
and that was useful, because I know that
{{{1/4+1/3=3/12+4/12=7/12}}} so {{{pi/4+pi/3=7pi/12}}}
and everybody knows that
{{{sin(pi/4)=cos(pi/4)=sqrt(2)/2}}}
{{{sin(pi/3)=sqrt(3)/2}}} and
{{{cos(pi/3)=1/2}}}
So
{{{sin(7*pi/12)=sin(pi/4+pi/3)=sin(pi/4)cos(pi/3)+cos(pi/4)sin(pi/3)}}}=
{{{(sqrt(2)/2)(1/2)+(sqrt(2)/2)(sqrt(3)/2)=sqrt(2)/4+sqrt(2)sqrt(3)/4 = sqrt(2)/4+sqrt(6)/4=(sqrt(2)+sqrt(6))/4}}}
BACK TO THE PROBLEM
I am going to use {{{highlight(sin(7*pi/12)=(sqrt(2)+sqrt(6))/2))}}}
However, it turns out that all the answers are either that expression, or (-1) times that, so if you were meant to use the fishy expression, you'll easily figure out the intended answers
a) {{{sin(-anything)=-sin(anything)}}} so
{{{sin(-7pi/12)=-sin(7pi/12)=highlight(-((sqrt(2)+sqrt(6))/4))}}}
b) {{{5pi/12+7pi/12=12pi/12=pi}}} so {{{5pi/12}}} and {{{7pi/12}}} are supplementary angles. They add up to {{{pi}}}, which is {{{180^o}}}.
And we know that {{{sin(A)sin(pi-A)}}}
so {{{sin(5pi/12)=sin(7pi/12)=(sqrt(2)+sqrt(6))/2)}}}
and {{{sin(-5pi/12)=-sin(-pi/12)=highlight(-((sqrt(2)+sqrt(6))/2)))}}}
c) {{{pi/12=7pi/12-6pi/12=7pi/12-pi/2}}}
I think we are expected to go to that table of trigonometric identities to find
{{{cos (A-B)=cos(A)cos(B)+sin(A)sin(B)}}}
Luckily, as everybody knows, {{{cos(pi/2)=0}}} and {{{sin(pi/2)=1}}}
So {{{cos(pi/12)=cos(7pi/12-pi/2)=cos(7pi/12)cos(pi/2)+sin(7pi/12)sin(pi/2) = cos(7pi/12)*0+sin(7pi/12)*1=sin(7pi/12)=highlight((sqrt(2)+sqrt(6))/2))}}}