= Solution
For $\lambda>0$, the <scalar potential> is minimized at $\phi^\dagger\phi=v^2/2$. With the specified upper-component <vacuum expectation value> $\phi_0=(v,0)^T/\sqrt2$ and <hypercharge> $Y=-I/2$, the unbroken generator is
$$
Q=T^3+Y=\begin{pmatrix}0&0\\0&-1\end{pmatrix},\qquad Q\phi_0=0.
$$
The other three independent generator directions act nontrivially on $\phi_0$. Thus the connected unbroken gauge subgroup is $U(1)_{\rm em}$, while three gauge directions are broken. The <normalization> here is $Q=T^3+Y$, not the alternative convention in which <hypercharge> is doubled.
Apply the <covariant derivative> directly to the vacuum:
$$
D_\mu\phi_0=-\frac{iv}{2\sqrt2}
\begin{pmatrix}g_2A_\mu^3-g_1B_\mu\\g_2(A_\mu^1+iA_\mu^2)\end{pmatrix}.
$$
In the mostly-plus convention the canonical <scalar> kinetic term is $-(D_\mu\phi)^\dagger D^\mu\phi$. Its quadratic vector terms are
$$
\mathcal L_{\rm mass}=-\frac{v^2}{8}
\left[g_2^2(A_\mu^1A^{1\mu}+A_\mu^2A^{2\mu})
+(g_2A_\mu^3-g_1B_\mu)(g_2A^{3\mu}-g_1B^\mu)\right].
$$
Define $s_W=\sin\theta_W=g_1/\sqrt{g_1^2+g_2^2}$ and $c_W=\cos\theta_W=g_2/\sqrt{g_1^2+g_2^2}$. Diagonalization gives
$$
\boxed{W_\mu^\pm=\frac{A_\mu^1\mp iA_\mu^2}{\sqrt2},\qquad
Z_\mu=c_WA_\mu^3-s_WB_\mu,\qquad
A_\mu^\gamma=s_WA_\mu^3+c_WB_\mu.}
$$
The neutral combination $Z$ lies along the nonzero <eigenvector> $(g_2,-g_1)$; the orthogonal combination $(g_1,g_2)$ is the massless <photon>. Comparing the <mass> terms with $-m_W^2W^+\cdot W^--m_Z^2Z^2/2$ yields
$$
\boxed{m_W=\frac{g_2v}{2},\qquad
m_Z=\frac v2\sqrt{g_1^2+g_2^2},\qquad
m_\gamma=0,\qquad\frac{m_W}{m_Z}=\cos\theta_W.}
$$
Finally insert the inverse neutral rotation into the coupling:
$$
g_2T^3A_\mu^3+g_1YB_\mu
=(g_2s_WT^3+g_1c_WY)A_\mu^\gamma
+(g_2c_WT^3-g_1s_WY)Z_\mu.
$$
Both coefficients in the <photon> term agree, giving
$$
\boxed{e=g_2\sin\theta_W=g_1\cos\theta_W
=\frac{g_1g_2}{\sqrt{g_1^2+g_2^2}},\qquad D_\mu\supset-ieQA_\mu^\gamma.}
$$
The upper vacuum component is electrically neutral and the lower <scalar> component has charge $-e$, consistently with the source's negative <hypercharge>. This doublet is the conjugate convention to the commonly displayed positive-hypercharge doublet; it does not change the vector masses or the electromagnetic coupling. The four real <scalar> components supply three would-be <Goldstone modes> and one physical radial Higgs mode, preserving the total physical degrees of freedom.
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