Remember that $\sin X = \dfrac{\text{opposite to }X}{\text{hypotenuse}}$. Which sides are opposite to $X$ and the hypotenuse? Assign them lengths in the same ratio $4:7$.

Once you know the hypotenuse and one leg, use $a^2 + b^2 = c^2$ to find the other leg. Then, think about which sides are opposite and adjacent to angle $Y$ to compute $\tan Y$.

In Desmos, type the expression sqrt(1-(4/7)^2)/(4/7). This uses $\cos X = \sqrt{1-\sin^2 X}$ and $\tan Y = \dfrac{\cos X}{\sin X}$ (since $Y$ is complementary to $X$). The value Desmos gives is the numerical value of $\tan Y$.

In separate lines, type each option exactly as written: sqrt(33)/4, 4/sqrt(33), sqrt(33)/7, 7/sqrt(33). Compare their decimal values to the decimal from step 1; the choice whose value matches is the correct answer.

Angle $Z$ is the right angle, so side $XY$ is the hypotenuse. The legs are $XZ$ and $YZ$.

Relative to angle $X$:

• The opposite side is $YZ$.
• The adjacent leg (not the hypotenuse) is $XZ$.
• The hypotenuse is $XY$.

Relative to angle $Y$:

• The opposite side is $XZ$.
• The adjacent leg is $YZ$.
• The hypotenuse is still $XY$.

By definition, for any angle in a right triangle,

We are told $\sin X = \dfrac{4}{7}$, so

That means we can choose

• $YZ$ (opposite $X$) $= 4$,
• $XY$ (hypotenuse) $= 7$.

Any similar triangle scaled up or down would keep the same ratios, so using $4$ and $7$ is valid.

Now we know two sides of the right triangle:

• One leg: $YZ = 4$,
• Hypotenuse: $XY = 7$.

Let the remaining leg be $XZ$. By the Pythagorean theorem,

Substitute the values:

So $XZ = \sqrt{33}$ (we take the positive root because a side length must be positive).

For angle $Y$, tangent is defined as

From our labels:

• Opposite to $Y$ is $XZ = \sqrt{33}$,
• Adjacent to $Y$ is $YZ = 4$.

So

This matches choice A) $\dfrac{\sqrt{33}}{4}$.