Find the exact value of sin(−7pi/12):
So, the way I'd do this is using the half angle trig identity for Sin. So that turns
$\displaystyle sin\frac{-7\pi}{12}$
intp
$\displaystyle sin\frac{-7\pi}{6}$
the half angle identity lets you stuff that into
$\displaystyle \pm \sqrt{\frac{1-cos\frac{-7\pi}{6}}{2}}$
Can you solve from there?
ok....so do you know how to turn $\displaystyle cos\frac{-7\pi}{6}$ into an exact answer?
Assuming you do, the answer is $\displaystyle \frac{-\sqrt{3}}{2}$
so plugging that into the equation, you get
$\displaystyle \pm\sqrt{\frac{1+\frac{\sqrt{3}}{2}}{2}}$
simplifying, you get
$\displaystyle \pm\sqrt{\frac{\frac{2+\sqrt{3}}{2}}{2}}$
that then get simplified further to
$\displaystyle \pm\sqrt{\frac{2+\sqrt{3}}{4}}$
and finally
$\displaystyle \pm\frac{\sqrt{2+\sqrt{3}}}{2}$
the initial equation is $\displaystyle sin\frac{-7\pi}{12}$, which puts it in 3rd quadrant, so we know it needs to be negative, or
$\displaystyle -\frac{\sqrt{2+\sqrt{3}}}{2}$
Is it clearer?
Alternatively, note that
$\displaystyle \sin \frac {-7 \pi}{12} = - \sin \frac {7 \pi}{12}$
$\displaystyle = - sin \left( \frac {4 \pi}{12} + \frac {3 \pi}{12} \right)$
$\displaystyle = - sin \left( \frac {\pi}3 + \frac {\pi}4 \right)$
Now apply the addition formula for sine, and don't forget that you have a negative sign out in front...
that's the most simple exact answer. You could actually solve it and get a decimal value, but it wouldn't be exact any more since you'd have to round it. The (inaccurate) decimal answer is like -0.9659258263.... but if this is for class and your professor/teacher is anything like mine, giving a decimal answer would result in a fail.
*edit* Good call Jhevon. There are certainly other ways to solve this equation.