Displacement of a Pendulum to find Work done by Force

[shinify]

I'm having trouble with this problem. I did: F_Tcosθ-mg= 0, solving for F_T, getting F_T= mg/cosθ. Then, along the x-axis, F-F_Tsinθ=0, solving for F, getting F= mgtanθ. Not sure how to go about it from here. What is the direction of the displacement?

 

[haruspex]

The force is always horizontal, so for the work done you want the horizontal displacement. But.. why not just use conservation of energy?

 

[shinify]

What do you mean?

 

[haruspex]

The force has done work on the system. Where has that work gone? There are three possibilities: PE, KE and frictional losses. Which apply here?
If you want to do it by integration, ∫F.ds, since the force acts horizontally that becomes ∫F.dx, where F is the scalar magnitude of F and dx is the horizontal component of ds. You have F as a function of θ, so next you need dx in terms of θ and dθ.