What Angle Minimizes Force to Make a Block Slip?

[atarr3]

Homework Statement

A person pulls on a block with a force F, at an angle theta with respect to the horizontal. The coefficient of friction between the block and the ground is mu. For what theta is the F required to make the block slip a minimum.

Homework Equations

Net force equations for static objects

The Attempt at a Solution

So I'm pretty sure I've got all of my equations right, but I'm having trouble simplifying my results further.
$$Ff=\mu N$$ when the box slips and $$N=Mg-Fsin\theta$$ Substitution gives $$\mu\left(Mg-Fsin\theta\right)=Fcos\theta$$ I don't know how to solve for theta.

 

[aim1732]

For theta for minimum F, write F as an explicit function of theta and differentiate w.r.t theta. Put it equal to zero(the calculus concept of maxima and minima).

 

[atarr3]

But I'm not trying to find the minimum F, I'm trying to find the minimum theta.

 

[atarr3]

Ok I think I have it now. I maximize the denominator in $$F=\frac{\mu Mg}{cos\theta+\mu sin\theta}$$ so the derivative is $$\mu cos\theta-sin\theta=0$$ So $$tan\theta=\mu$$

 

[aim1732]

As an afterthought, don't you think the minimum theta is zero?