What is the Minimum Angle for a Ladder to Not Slip Against a Wall?

[shards5]

Homework Statement

So the original problem goes as the following.

A uniform ladder of mass (m) and length (L) leans against a frictionless wall, see figure. If the coefficient of static friction between the ladder and the ground is 0.41, what is the minimum angle (q) between the ladder and the floor at which the ladder will not slip?

And I get the whole concept behind how to do it which is basically using sum of the forces and sum of the torques but my math seems to be wrong and different from the solutions I found online (my answer was 0.014 degrees which is obviously wrong). So can you tell me where I messed up in my math?

Homework Equations

The Attempt at a Solution

Sum of the forces = 0
N_w = Normal force done on the ladder by the wall
N_f = Normal force done on the ladder by the floor
cg = Center of gravity (bad habit of mine that I use to distinguish between my other mg)

x: N_w - F_s = ma (a = 0 thus ma = 0)
y: N_f - cg = 0

Thus:
N_f = cg
F_s = N_w

Sum of the Torques = 0
Set the axis of rotation at the base of the ladder
N_w sin(theta) - (g sin(theta))/2 = 0
Substituting Nw for Friction static and then substituting Friction static with coefficient of static friction times cg I get the following:
uk cg sin (theta) - (cg cos(theta))/2 = 0
Canceling out the cg I get the following:
uk sin (theta) - (cos(theta))/2 = 0
Now this is where I screw up so can someone explain how I screwed up.
Moved cos to the other side
uk sin(theta) = (cos(theta))/2
Multiplied 2 to get rid of the 1/2 on the right side
2uk sin(theta) = cos(theta)
Divided by sin (theta)
2 uk = cos(theta)/sin(theta)
Which becomes
2 uk = arctan(theta)
Thus
theta = tan(2uk)


Thanks for your help and sorry for asking for putting out so much detail for something so simple.

 

[Doc Al]

2 uk = cos(theta)/sin(theta)

This is good. (Really, μ_s, since it's static friction.)

Which becomes
2 uk = arctan(theta)

Careful: cos/sin = cot, not tan.

Thus
theta = tan(2uk)

(1) Fix the mistake I pointed out regarding cot vs. tan.
(2) Make sure your calculator is set to degree mode and not radians.