reflected electric and magnetic fields component
I took your picture from your other question and I edited to make it match to your problem. 



As you can see the wave progresses in the [3,0,4] direction. This is the argument of the complex exponential. Remember than ax+by+cz=K (K is a constant) would be a plane and the vector perpendicular to it is [a,b,c].

Vector \$\vec \beta _i\$ is therefore [3,0,4]. You can see from the picture that the reflected \$\vec \beta _r\$ is [3,0,-4]. It reflects on the z=0 plane (and that's why z changes sign).

In the other hand, the cos and sin components allow to write the vector from its polar coordinates.

If you look at \$\vec H _i\$ it is easy to verify that its angle with \$\beta _z\$ is \$\theta_i\$ therefore its coordinates are \$[-H_i  \cos(\theta_i) , 0, H_i sin(\theta_i)]\$.

Because \$\Gamma\$ is negative we have a direction change for the reflected E field. Also E, H and propagation direction must follow the 3 fingers rule, before and after the reflection, as they do in the figure.

For \$\vec H _r\$ you can see from the figure that its coordinates will be \$[-H_r  \cos(\theta_r) , 0, -H_r sin(\theta_r)]\$ as in the solution.