$ \text{Step 1: Given Data} $
Given that the angle $\theta = 60^\circ$, and the force $F = 500 \, \text{N}$, we are required to estimate the magnitudes of the two components of $F$ directed along the axes of $AB$ and $AC$.

$ \text{Step 2: Free Body Diagram for Link AB} $
To proceed, we analyze the free body diagram for link $AB$ to calculate the component of force.

$ \text{Step 3 : Calculate the Magnitude of Force } F_{AB} $
To find the magnitude of force $F_{AB}$, we will use the law of sines:

\begin{array}{c}
\frac{{F_{AB}}}{{\sin 60^\circ}} = \frac{F}{{\sin 75^\circ}}
\end{array}

$ \text{Step 4: Substitute Known Values} $
Substituting the known values into the equation yields:

\begin{array}{c}
\frac{{F_{AB}}}{{\sin 60^\circ}} = \frac{{500 \, \text{N}}}{{\sin 75^\circ}} \\
F_{AB} = 500 \, \text{N} \times \frac{\sin 60^\circ}{\sin 75^\circ} \\
F_{AB} \approx 448.28 \, \text{N}
\end{array}

$ \text{Step 5: Free Body Diagram for Link AC} $
Next, we analyze the free body diagram for link $AC$ to calculate the component of force.

$ \text{Step 6: Calculate the Magnitude of Force } F_{AC} $
To find the magnitude of force $F_{AC}$, we will use the law of sines:

\begin{array}{c}
\frac{{F_{AC}}}{{\sin 45^\circ}} = \frac{F}{{\sin 75^\circ}}
\end{array}

$ \text{Step 7: Substitute Known Values} $
Substituting the known values into the equation yields:

\begin{array}{c}
\frac{{F_{AC}}}{{\sin 45^\circ}} = \frac{{500 \, \text{N}}}{{\sin 75^\circ}} \\
F_{AC} = 500 \, \text{N} \times \frac{{\sin 45^\circ}}{{\sin 75^\circ}} \\
F_{AC} \approx 366.02 \, \text{N}
\end{array}