Question 576·Medium·Math

Since $Z$ is the right angle, $XY$ is the hypotenuse, the side across from it. From $X$, $YZ$ is the opposite leg, the side across from $X$. Sine means opposite over hypotenuse, so $\sin X=\frac{YZ}{XY}=\frac47$.

The fraction tells you that $YZ$ and $XY$ are in a $4$-to-$7$ ratio. Use a scaled copy with $YZ=4$ and $XY=7$. Any other matching triangle scales both lengths equally, so its trig ratios are the same.

The Pythagorean theorem says the squares of the two legs add to the square of the hypotenuse. Here $XZ$ is the missing leg, and $XY=7$ is the hypotenuse, so $XZ^2+4^2=7^2$.

Subtract $4^2$ from both sides: $XZ^2=7^2-4^2$. Type $7^2-4^2$ in Desmos. It shows $33$, which is $XZ^2$, not $XZ$.

Take the positive square root because a side length can't be negative: $XZ=\sqrt{33}$. Keep the radical because the choices give exact values.

From $Y$, $XZ$ is opposite and $YZ$ is adjacent. Tangent means opposite over adjacent, so $\tan Y=\frac{XZ}{YZ}=\frac{\sqrt{33}}{4}$. Switching acute angles swaps the legs' opposite and adjacent roles. Choice D.