= Solution
Take $m,k,\gamma,F_0>0$ and $\omega>0$, and write $D_\omega=k-m\omega^2$ and $B_\omega=m\gamma\omega$. Substitution of $z=A\cos\omega t$ into the <forced harmonic oscillator> equation gives
$$
A D_\omega\cos\omega t-A B_\omega\sin\omega t
=F_0\cos\phi_0\cos\omega t-F_0\sin\phi_0\sin\omega t.
$$
Equating the independent <sine> and <cosine> coefficients yields
$$
\boxed{A D_\omega=F_0\cos\phi_0,\qquad A B_\omega=F_0\sin\phi_0.}
$$
Thus the required <phase matching for a displacement referenced driven oscillator> is
$$
\boxed{(k-m\omega^2)\sin\phi_0=m\gamma\omega\cos\phi_0.}
$$
For a positive <amplitude> $A$, one must also choose the correct quadrant: $\cos\phi_0=D_\omega/\sqrt{D_\omega^2+B_\omega^2}$ and $\sin\phi_0=B_\omega/\sqrt{D_\omega^2+B_\omega^2}$. Squaring and adding the coefficient equations gives
$$
\boxed{A=\frac{F_0}{\sqrt{(k-m\omega^2)^2+(m\gamma\omega)^2}}.}
$$
For arbitrary forcing <phase>, the steady response is instead $z=A\cos(\omega t+\phi_0-\delta)$, where $\delta$ has the sine and cosine just specified. A change of time origin can put this displacement into $A\cos\omega t$, but it also changes the stated forcing <phase>.
The source's requested parameter list for $A$ omits $\gamma$; in general it cannot be omitted. As a concrete counterexample, take $m=k=F_0=\omega=1$ and $\phi_0=\pi/2$. The prescribed cosine response exists for every $\gamma>0$, but $A=1/\gamma$, so fixing all four listed parameters does not fix the <amplitude>. Likewise, with fixed $\phi_0=0$, positive damping and nonzero $\omega$, the second coefficient equation prevents a nonzero pure-cosine response. The forcing <phase> is an essential part of the condition.
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