Equation involving cosine I don't really know how to crack this one. Any help appreciated. $$\cos\left(d\sqrt{4-d^2}\right)=-\frac{d}2$$ Why is MSE forcing me to write more? Jeez.

As Sabyasachi wrote, the domain is $d\in [-2,2]$. Obviously $d=-2$ is a solution but there  is another one between $-2$ and $0$ which cannot be expressed analytically (I suppose). So the solution must be found using numerical methods such as Newton, provided a reasonable starting point. By inspection, the value of $$f(d)=\cos\left(d\sqrt{4-d^2}\right)+\frac{d}2$$ is negative for $d=-1$ and positive for $d=0$.  
So, start iterating at $d=- \frac {1} {2}$ and update $d$ according to $$d_{new}=d_{old}-\frac {f(d_{old})}{f'(d_{old})}$$ The successive iterates on Newton method will then be $-0.659241$, $-0.654719$ which is the solution for six significant digits.
