I have a problem where I have an initial range, bearing, and elevation angle of an aircraft. I want to calculate the $x$, $y$ and $z$ coordinates every second. Is there any formula to do so?
Initial course of the aircraft is 60 deg and speed is 245 m/s.
As an example of how this could work, I'll assume that the $z$ axis points up, $y$ points north, and $x$ points east. I'll also assume that course and bearing are both measured clockwise from north, that is, north is zero degrees and east is $90$ degrees, and that a positive elevation angle is above horizontal. I'll assume that range is measured in a straight line through space, that is, it is not just the horizontal distance but includes the vertical component. I'll also assume the aircraft is flying straight and level (not turning, climbing, or descending).
If any of those assumptions differ from the way your coordinates and angles are set up, the following formulas will have to be adjusted.
Let the initial range, bearing, and elevation of the aircraft be $r_0,$ $\theta_0,$ and $\phi_0.$ Let $P_0$ be the initial position of the aircraft and $Q_0$ the point at the same $x,y$ coordinates in the $x,y$ plane. Then the origin $O$ and the points $P_0$ and $Q_0$ are vertices of a right triangle with angle $\phi_0$ at $O$ and hypotenuse $r.$ The opposite leg $P_0Q_0$ gives the initial altitude $z_0,$ and the adjacent leg $OQ_0$ gives the horizontal distance to the initial point, which would equal the initial $y$ coordinate, $y_0,$ if $\theta_0 = 0$ but would be the initial $x$ coordinate, $x_0,$ if $\theta_0 = 90^\circ.$ \begin{align} z_0 &= r_0 \sin\phi_0 \\ y_0 &= r_0 \cos\phi_0 \cos\theta_0 \\ x_0 &= r_0 \cos\phi_0 \sin\theta_0 \end{align}
Given a course angle $\psi$ and speed $v$ meters per second, the velocity vector has components \begin{align} v_x &= v \sin\psi \\ v_y &= v \cos\psi \\ v_z &= 0 \end{align} in the $x,$ $y,$ and $z$ directions, respectively. The position after $t$ seconds is \begin{align} x(t) &= x_0 + v_x t \\ y(t) &= y_0 + v_y t \\ z(t) &= z_0 + v_z t. \end{align}