Bounds of Integration for Random Oriented particle

[relskhan]

In the Stoner-Wohlfarth model, a uniaxial, non-interacting particle is cooled to very low temperature with no exposure to an external field. Therefore, the orientation of each particle is random, if you have a group of particles. In their paper, they integrate such that:
$$\langle \cos (\Theta )\rangle =\int_0^{\frac{\pi }{2}} \sin (\Theta ) \cos (\Theta ) \, d\Theta$$

I am having a hard time understanding why they only integrate from 0 to pi over two, instead of 0 to pi. Can anyone shine any enlightenment on this?

 

[fzero]

A copy of the paper can be found at http://spin.nanophys.kth.se/spin/stoner-wohlfarth.pdf There's a lot of discussion around Fig. 4 where they talk about how the symmetries of the problem allow them to reproduce the solutions everywhere in parameter space from the region $0 \leq \theta,\phi \leq \pi/2$.

 

[relskhan]

A copy of the paper can be found at http://spin.nanophys.kth.se/spin/stoner-wohlfarth.pdf There's a lot of discussion around Fig. 4 where they talk about how the symmetries of the problem allow them to reproduce the solutions everywhere in parameter space from the region $0 \leq \theta,\phi \leq \pi/2$.

That helped me tremendously - thank you!