R S = 20 S T = 48 T R = 52 The side lengths of

Choice B is correct. It's given that right triangle $RST$ is similar to triangle $UVW$, where $S$ corresponds to $V$ and $T$ corresponds to $W$. It's given that the side lengths of the right triangle $RST$ are $RS=20$, $ST=48$, and $TR=52$. Corresponding angles in similar triangles are equal. It follows that the measure of angle $T$ is equal to the measure of angle $W$. The hypotenuse of a right triangle is the longest side. It follows that the hypotenuse of triangle $RST$ is side $TR$. The hypotenuse of a right triangle is the side opposite the right angle. Therefore, angle $S$ is a right angle. The adjacent side of an acute angle in a right triangle is the side closest to the angle that is not the hypotenuse. It follows that the adjacent side of angle $T$ is side $ST$. The opposite side of an acute angle in a right triangle is the side across from the acute angle. It follows that the opposite side of angle $T$ is side $RS$. The tangent of an acute angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side. Therefore, $\tan T=\frac{RS}{ST}$. Substituting $20$ for $RS$ and $48$ for $ST$ in this equation yields $\tan T=\frac{20}{48}$, or $\tan T=\frac{5}{12}$. The tangents of two acute angles with equal measures are equal. Since the measure of angle $T$ is equal to the measure of angle $W$, it follows that $\tan T=\tan W$. Substituting $\frac{5}{12}$ for $\tan T$ in this equation yields $\frac{5}{12}=\tan W$. Therefore, the value of $\tan W$ is $\frac{5}{12}$.

Choice A is incorrect. This is the value of $\sin W$.

Choice C is incorrect. This is the value of $\cos W$.

Choice D is incorrect. This is the value of $\frac{1}{\tan W}$.