For an engineering problem I am trying to obtain the polar plot of an 'antenna factor' function by hand. The function for the array factor is given as:
$$f(\Psi) = \frac{sin(N\frac{\Psi}{2})}{Nsin(\frac{\Psi}{2})}$$
Where $\Psi = \beta dcos(\theta) + \alpha$
For this particular problem $N=5$, and $d=0.35\lambda$. I worked out $\alpha=0.7\pi$ and $\beta d = \frac{2 \pi}{\lambda}\times0.35\lambda = 0.7 \pi$
Substituting $\Psi = 0.7\pi cos(\theta) + 0.7 \pi$ into the formula for the array factor gives
$$f(\theta) = \frac{sin(5\frac{0.7\pi cos(\theta) + 0.7 \pi}{2})}{5sin(\frac{0.7\pi cos(\theta) + 0.7 \pi}{2})}$$
I am not allowed to use any computing tools for this exercise and so I am trying to plot the function by hand. When I substitute values for $\theta$ I don't get what I expect, the values that result are always close to 1, such as 0.994 for $\theta =0^\circ$ and $0.994$ for $\theta = 30^\circ$. The array factor for this problem should look as the polar plot below. Could someone please help me with how to obtain the polar plot for this function given the formula above.
