This will surely be a stupid question, but it bugs me.
Let's consider an inclined plane with friction.
We all know that the friction force is given by
$$f = \mu N$$
Where $N = mg\cos\theta$, the normal force.
Now, when I have to deal with non conservation of energy, I set $\mathcal{L} = -\Delta E$, but here is my doubt.
$\mathcal{L}$ is the work done by the friction force, and by definition we have
$$\mathcal{L} = f\cdot s$$
Where $s$ is the displacement. Now the dot $\cdot$ means "time cosine", but the force $f$ itself is $\mu N$ which has a cosine within. Hence
$$\mathcal{L} = \mu m g s \cos^2\theta$$
But it's wrong, since we know it to be
$$\mathcal{L} = \mu m g s \cos\theta$$
Where is the other cosine?
Thank you and sorry for this stupid question.