= Solution
Assume an initially unfractured homogeneous <isotropic> material described by a linear <Mohr-Coulomb failure criterion>, with the given <coefficient of friction> $\mu_f=0.5$ representing the relevant <internal friction angle>. Use spatially uniform <pore fluid pressure>, <effective stresses> $\sigma_i'=\sigma_i-P_f$, and compression-positive signs. Then
$$
|\tau|=c+\mu_f\sigma_n',\qquad\varphi=\arctan\mu_f=26.565^\circ.
$$
The <Mohr circle> in the $\sigma_1',\sigma_3'$ plane has centre $C=(\sigma_1'+\sigma_3')/2$ and radius $R=(\sigma_1'-\sigma_3')/2$. At first failure it is tangent to the <failure envelope>. The radius to the positive-shear tangent point is perpendicular to the line of slope $\tan\varphi$, hence makes angle $90^\circ+\varphi$ with the positive normal-stress axis.
If $\theta$ is the angle between the \b[fault normal] and $\sigma_1$, the <stress transformation> is
$$
\sigma_n'=C+R\cos2\theta,\qquad\tau=R\sin2\theta.
$$
Thus $2\theta=90^\circ+\varphi$, $\theta=58.2825^\circ$, and the angle of the fault itself to horizontal $\sigma_1$ is
$$
\boxed{\delta=90^\circ-\theta=45^\circ-\frac\varphi2=31.7175^\circ.}
$$
The two conjugate <thrust faults> therefore strike approximately east–west and dip about \b[$32^\circ$ north and $32^\circ$ south]. This is the <fault dip>, not the $58.3^\circ$ fault-normal angle. Both <sediments> and basement have the same optimum orientation under these assumptions because their stipulated friction slopes are equal; their <cohesion> and absolute failure <stresses> could still differ.
The first two panels in <3/b/solution> give the labeled construction. Constant <cohesion> shifts the straight <failure envelope> without changing its slope, and uniform <pore fluid pressure> translates the <Mohr circle> in normal-stress coordinates; neither changes this tangency angle. In the illustrative cohesionless case, $R/C=\sin\varphi$, giving $\sigma_1'/\sigma_3'=(1+\sin\varphi)/(1-\sin\varphi)\simeq2.618$. Pre-existing weaknesses, a nonuniform <stress> field, anisotropic strength, or a nonlinear <failure envelope> invalidate the simple optimum-orientation prediction.<Stress and strength lecture notes by MIT OpenCourseWare>[Lecture 10 develops the traction, pore-pressure and Mohr-circle framework; the signs and angle definitions above specify the convention used here.]{ref}
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