One page 159 of The Arithmetic of Hyperbolic Manifolds by Maclachlan and Reid:-

In more detail, suppose that $\Gamma$ is a $(l,m,n)$-triangle group where $1/l+1/m+1/n<1$ so that $\Gamma$ has the presentation

 

$\langle x,y\mid x^l=y^m=(xy)^n=1 \rangle$.

 

Then $\mathbb{Q}(\mathrm{tr}\Gamma)=\mathbb{Q}(\cos\pi/l,\cos\pi/m,\cos\pi/n)$ (see (3.25)), and the invariant trace field is a subfield of this totally real number field (see Exercise 4.9, No. 1).

More generally, Maclachlan and Reid should have the answers to all your questions about trace fields.

One page 159 of The Arithmetic of Hyperbolic Manifolds by Maclachlan and Reid:-

In more detail, suppose that $\Gamma$ is a $(l,m,n)$-triangle group where $1/l+1/m+1/n<1$ so that $\Gamma$ has the presentation

$\langle x,y\mid x^l=y^m=(xy)^n=1 \rangle$.

Then $\mathbb{Q}(\mathrm{tr}\Gamma)=\mathbb{Q}(\cos\pi/l,\cos\pi/m,\cos\pi/n)$ (see (3.25)), and the invariant trace field is a subfield of this totally real number field (see Exercise 4.9, No. 1).

More generally, Maclachlan and Reid should have the answers to all your questions about trace fields.