Finding minimum value of force P given static friction

[Zipzap]

Homework Statement

A crate of weight Fg is pushed by a force P on a horizontal
floor. (a) If the coefficient of static friction is us and P
is directed at angle theta below the horizontal, show that the
minimum value of P that will move the crate is given by

P = ( us*Fg*sec(theta)) / (1 - us*tan(theta))


(b) Find the minimum value of P that can produce motion
when us " 0.400

Homework Equations

The Attempt at a Solution

All right, I managed to do (a) on my own, but I'm dead stuck on (b). I've only gotten as far as inputting us = 0.400 in, and all I know is that I need to find an angle such that I get a minimum for P. I'm unsure as to how to solve this problem without inputting a bunch of angles and using trial-and-error to get my answer. Can anyone help me?

 

[Pagan Harpoon]

Differentiate your expression for P with respect to Theta and set it to 0, find the minimum in the usual way. I think that will work.

 

[Zipzap]

I tried that, and then I got some insane result that I have no idea how to solve:

http://www.wolframalpha.com/input/?i=derivative+of+(0.4secx)+/+(1-0.4tanx)

That's what I get, and I'm once again stuck from here...

 

[Pagan Harpoon]

You will need to solve that equation. It's not as bad as it looks. Straight away you can simplify it by multiplying top and bottom of the second fraction by 1-0.4tan(x), then the denominators for both are the same, so you can throw them away and you have:

$$\frac{0.16}{cos^{3}(x)}+\frac{0.4sin(x)}{cos^{2}(x)}-\frac{0.16sin^{2}(x)}{cos^{3}(x)}=0$$

You can solve this to get the root you want in a simpler form than it is given on the WolframAlpha page.

Keep in mind that 1-sin^2(x)=cos^2(x), this property is useful in the process of solving the equation.