Find The Coordinates of Atoms I have this problem that I have not been able to solve for a very long time.
I have 4 atoms in 2D space (XY). I know the coordinates of the blue atoms and the distances between all atoms (blue and red).
How can I find the coordinates of the red atoms?

 A: Polar coordinates will be very useful here.
To find the location of the carbon atom, imagine that the origin of a polar coordinate system is at the left-most of the two calcium atoms. In that case, the carbon atom is at the point $(r_C,\theta_C)=(2.4,0.222)$ in that polar coordinate system (angle converted to radians). This means that in a rectangular coordinate system centered on the left calcium atom, the carbon atom's coordinates are given by 
\begin{align}
x_C&=r_C\cos\theta_C=2.4\cos(0.222)=2.34 \\
y_C&=r_C\sin\theta_C=2.4\sin(0.222)=0.53
\end{align}
Switching back to the coordinate system shown in the picture, we now know that the carbon atom is $2.34$ units to the right of the left calcium atom and $0.53$ units above it. We can add those relative coordinates to the coordinates of the left calcium atom as shown in the picture to find that the carbon atom is located at the point $(1.0+2.34,2.0+0.53)=(3.34,2.53)$.

We can use the exact same trick to find the coordinates of the nitrogen atom. In a polar coordinate system centered on the left calcium atom, the nitrogen atom is at the point $(r_N,\theta_N)=(1.5,-0.262)$ (angle converted to radians again; also, this angle happens to be exactly $-\frac{\pi}{12}$, but the decimal approximation is perfectly fine). Therefore, the nitrogen atom's coordinates in a rectangular coordinate system centered on the left calcium atom are:
\begin{align}
x_N&=r_N\cos\theta_N=1.45 \\
y_N&=r_N\sin\theta_N=-0.39
\end{align}
Switching back to the coordinate system shown in the picture again, we can add the relative coordinates of the nitrogen atom to the displayed coordinates of the left calcium atom to find that the nitrogen atom is located at the point $(1.0+1.45,2.0-0.39)=(2.45,1.61)$.
As a quick sanity check, performing the same kinds of calculations relative to the right-most calcium atom yields very similar coordinates for the carbon and nitrogen atoms, so we can be confident that the above results are reasonably correct.
