To derive these fictitious forces, it's helpful to be able to convert between the coordinates ( x ′ , y ′ , z ′ ) of the rotating reference frame and the coordinates ( x , y , z ) of an inertial reference frame with the same origin. If the rotation is about the z axis with a constant angular velocity Ω Ω (so z ′ = z and d θ θ d t ≡ ≡ Ω Ω , which implies θ θ ( t ) = Ω Ω t + θ θ 0 for some constant θ θ 0 where θ θ ( t ) denotes the angle in the x − − y -plane formed at time t by ( x ′ , y ′ ) and the x -axis), and if the two reference frames coincide at time t = 0 (meaning ( x ′ , y ′ , z ′ ) = ( x , y , z ) when t = 0 , so take θ θ 0 = 0 or some other integer multiple of 2 π π ), the transformation from rotating coordinates to inertial coordinates can be written x = x ′ cos ⁡ ⁡ ( θ θ ( t ) ) − − y ′ sin ⁡ ⁡ ( θ θ ( t ) ) y = x ′ sin ⁡ ⁡ ( θ θ ( t ) ) + y ′ cos ⁡ ⁡ ( θ θ ( t ) ) whereas the reverse transformation is x ′ = x cos ⁡ ⁡ ( − − θ θ ( t ) ) − − y sin ⁡ ⁡ ( − − θ θ ( t ) ) y ′ = x sin ⁡ ⁡ ( − − θ θ ( t ) ) + y cos ⁡ ⁡ ( − − θ θ ( t ) ) .