Members of a truss are pin connected at joint O

The members of a truss are pin connected at joint O. Determine the magnitudes of F_1 and F_2 for equilibrium. Set ϴ = 60°.

Image from: Hibbeler, R. C., S. C. Fan, Kai Beng. Yap, and Peter Schiavone. Statics: Mechanics for Engineers. Singapore: Pearson, 2013.

Solution:

Let us first draw a free body diagram and label our forces like so:

As stated in the question, we set ϴ = 60° in our diagram.

We will also pick which sides are positive. In this case, we will assume \rightarrow^+ is positive and \uparrow+ is positive.

Now, we will write each force in unit vector form as follows:

(Note the negative signs. When the component of a force is towards the negative quadrant, we set it as negative, since we assumed that \rightarrow^+ is positive and \uparrow+ is positive)

For the forces to be in equilibrium, the x and y (i and j in unit vector form) forces must add up to zero. Mathematically, we can write this as:

Let us now only look at the x forces (unit vector i):

Now, we will look at the y forces (unit vector j):

We can now solve both equations simultaneously which gives us:

This question can be found in Engineering Mechanics: Statics (SI edition), 13th edition, chapter 3, question 3-1