While the coefficient-bodies \(S_{k,n}=\{(a,b)\in {\mathbb{C}}^ 2:\) \(z+az^ k+bz^ n\in S\}\) of schlicht trinomials are completely known by the independent results of Rahman-Waniurski and Kasten-Schmieder (1979/80), this is not the case so far for certain subclasses such as close-to-convex or starlike trinomials. The present paper gives a complete description of the sets \(C_{k,n}=\{(a,b)\in {\mathbb{R}}^ 2:\quad z+az^ k+bz^ n\) is close-to-convex\(\}\) in the case k-1 \(| n-1\). Furthermore, it turns out that for k-1 \(| n-1\) there exists a strip \(| b| \leq b^*\), such that the properties of locally schlichtness, schlichtness, close-to-convexity and starlikeness are equivalent. The given bound is sharp, and a similar result holds for a certain strip \(| a| \leq a^*.\) In the case k-1 \(\nmid n-1\) the situation looks quite different.