  We apply recent circle tangency estimates due to Pramanik--Yang--Zahl to
prove sharp weighted Fourier extension estimates for the cone in $\mathbb{R}^3$
and $1$-dimensional weights. The idea of using circle tangency estimates to
study Fourier extension of the cone is originally due to Tom Wolff, who used it
in part to prove the first decoupling estimates. We make an improvement to the
best known Mizohata--Takeuchi-type estimates for the cone in $\mathbb{R}^3$ and
the $1$-dimensional weights as a corollary of our main theorem, where the
previously best known bound follows as a corollary of refined decoupling
estimates.
