= Solution
Set $\epsilon=v_a^2/c^2\ll1$. Expanding the <square root> in part (a) gives
$$
\sqrt{(1+\epsilon)^2-4\epsilon\cos^2\theta}
=1+\epsilon(1-2\cos^2\theta)+O(\epsilon^2).
$$
Hence, to first absolute order in $\epsilon$,
$$
v_f^2=c^2+v_a^2\sin^2\theta+O(v_a^4/c^2),\qquad
v_{\rm slow}^2=v_a^2\cos^2\theta+O(v_a^4/c^2).
$$
The <high-pressure limit of magnetosonic waves> can also be expressed as nonnegative speeds. Retaining the next squared-speed term gives the first relative correction for the slow mode:
$$
\boxed{v_f=c\left[1+\frac\epsilon2\sin^2\theta+O(\epsilon^2)\right],\qquad
v_{\rm slow}=v_a|\cos\theta|\left[1-\frac\epsilon2\sin^2\theta+O(\epsilon^2)\right]}.
$$
The exact <Alfvén speed> along the propagation direction remains $v_a|\cos\theta|$.
The <fast magnetosonic wave> is predominantly an <acoustic wave>, with <velocity> almost parallel to $\mathbf k$ and appreciable <mass density> and gas <pressure> variations. Its <magnetic pressure> correction is weak. The <Alfvén wave> is exactly incompressible, polarized perpendicular to the $\mathbf k$-$\mathbf B_0$ plane and restored by <magnetic tension>.
The <slow magnetosonic wave> is polarized inside that plane and almost perpendicular to $\mathbf k$. Indeed, the leading <sound speed> part of the matrix in (a) annihilates a vector perpendicular to $\mathbf k$, so only the smaller magnetic restoring force acts on that <MHD wave polarization>. Its compression is smaller by order $\epsilon$, with gas <pressure> changes almost cancelling <magnetic pressure> changes. It is often called the <pseudo-Alfvén wave> <MHD wave polarization> in this high <pressure> limit. Thus two distinct almost-incompressible magnetic <MHD wave polarizations> have the same leading speed, while the third is predominantly sound. For exactly perpendicular propagation the <Alfvén wave> and <slow magnetosonic wave> frequencies vanish; for parallel propagation the two transverse magnetic <MHD wave polarizations> are degenerate.
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