For angle $P$, identify which side is opposite $P$ and which side is the hypotenuse. Sine of an angle is defined as opposite over hypotenuse.

You know $PQ$ and $PR$, but the side opposite $P$ is $QR$. Use the Pythagorean theorem with $PQ$, $QR$, and $PR$ to find $QR$.

Once you know $QR$, form the ratio $\dfrac{\text{opposite}}{\text{hypotenuse}}$ relative to angle $P$ and simplify the fraction to match one of the choices.

In Desmos, type sqrt(15^2 - 9^2) to calculate the length of side $QR$ using the Pythagorean theorem. Note the numerical result.

In a new Desmos line, type (sqrt(15^2 - 9^2))/15 to represent $\dfrac{\text{opposite}}{\text{hypotenuse}} = \dfrac{QR}{PR}$. Compare the simplified value (or decimal) that Desmos shows with the answer choices.

In a right triangle, the hypotenuse is the side opposite the right angle.

Here, angle $Q$ is $90^\circ$, so side $PR$ is the hypotenuse.

• $PR = 15$ (hypotenuse)
• $PQ = 9$ (one leg)
• $QR$ is the other leg (unknown).

Trig ratios are defined relative to a specific angle (here, angle $P$):

• The hypotenuse is still $PR$.
• The side opposite angle $P$ is $QR$ (it does not touch $P$).
• The side adjacent to angle $P$ is $PQ$.

For sine, we will need the ratio $\sin(P) = \dfrac{\text{opposite}}{\text{hypotenuse}} = \dfrac{QR}{PR}$, so we must first find $QR$.},{

In a right triangle,

Subtract $81$ from both sides:

so

Now we know all needed sides relative to angle $P$:

• Opposite side: $QR = 12$
• Hypotenuse: $PR = 15$

So

This matches answer choice B.