We localize the zeros of complex harmonic trinomials \(r_\alpha (z)=z^k+\frac{\alpha }{\overline{z}^\ell }-1\) , where \(k, \ell \in {\mathbb {N}}\) , \(k>\ell \) with \(\gcd (k,\ell )=1\) . We prove that for \(|\alpha |<\left( \frac{\ell }{k+\ell }\right) ^\frac{\ell }{k}\left( \frac{k}{k+\ell }\right) \) , \(r_\alpha \) has k and \(\ell \) number of sense-preserving and sense-reversing zeros, respectively. However, \(r_\alpha \) has only \((k-\ell )\) sense-preserving zeros for \(|\alpha |>\left( \frac{\ell }{k-\ell }\right) ^\frac{\ell }{k}\left( \frac{k}{k-\ell }\right). \) For certain choices of the parameter \(\alpha \) , we obtain distinct annular sectors each containing exactly one non-singular zero of \(r_\alpha \) . Also, we provide zero-exclusion regions and pictorial demonstrations for certain \(r_\alpha \) .
