There is something I don't understand from a solved exercise of antennas coupling and I hope someone could help me to understand.

This is not an exercise I have to solve.

Given two dipoles each aligned on the z axis and placed at a distance \(0.1\lambda\) on the x axis so

\(\Psi = 0.1(2\pi) \sin(\theta)\cos(\varphi)\)

the array factor given by the example is:

\(F_A(\Psi) = 1 - 0.77\exp(\mathrm i(\Psi + 0.67))\)

and the maximum of radiation given by the exercise is

\(F_A(\Psi) = 1.09\) for \(\theta = \pi/2\) and \(\varphi = 0\).

The exercise explicitly says that the result is given calculating \(F_A(\Psi)\) for the angles just above.
But when I try to solve for \(\varphi\), I fall into a math error:

\(
\begin{align}
& 0.1(2\pi)\sin(\theta)\cos(\varphi) + 0.67 = 0\\
& \cos(\varphi) = -1.066
\end{align}
\)

since \(\arccos(.)\) is not defined for -1.066.
So what's the problem with my math ?

Thank you.

 

Perhaps I understand how the example comes to this solution ...

Given:

and its derivative is:

Now, equating it to 0, for definition we obtain the local maxima an minima in the range [0, π].
Obviously the derivative is 0 for φ = 0 and φ = π, so for φ = 0 we have the maxima, while the minima for φ = π..