How can I deduced the third point that forms a triangle in 3D. I have the cartesian location of two points say A, B and the lengths of the three sides.
I am mostly particular about the z-axis of this point because with trig functions, I have obtained perfect values for x,y axes. but the z-axis' value is a little bit erroneous.
Here is an image of the triangle:
On
Imagine the known points A and B are the endpoints of an axle, say from a bicycle wheel:
Think about the given lengths as the lengths of the spokes of the wheel. Knowledge about the lengths $\overline{AC}$ and $\overline{BC}$ alone will not give a single solution for the position of point C. In fact, there are infinitely many solutions on the rim of the wheel. For a bicycle wheel $\overline{AC} = \overline{BC}$, but even if they are different, the analogy holds: just imagine the spokes on one side to be longer than on the other.
So when you say you are looking for a point, you are actually looking for a circle.
Well, this question is a duplicate of your previous question "How to determine $x, y, z$ coordinates of the third vertex of a 3D triangle?". Isn't it clearly settled? You can ask this question as much as you want, but the answer is not going to change. The problem may not have a solution (due to a violation of one of the three triangle inequalities) and if there is one, it is not going to be unique but a whole circle of points $C$ that satisfy your conditions.