1. The problem is to understand why in step 5 of a certain derivation, the expression $bd = r \sin \theta$ holds. 2. Typically, in polar coordinates or trigonometric contexts, $r$ is the hypotenuse of a right triangle, $\theta$ is the angle, and $b$ and $d$ represent lengths related to the triangle. 3. The sine function is defined as the ratio of the length of the side opposite the angle to the hypotenuse: $$\sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}}.$$ 4. If $bd$ represents the length of the side opposite $\theta$, and $r$ is the hypotenuse, then by rearranging the sine definition, we get: $$\text{opposite side} = r \sin \theta.$$ 5. Therefore, $bd = r \sin \theta$ because $bd$ corresponds to the side opposite the angle $\theta$ in a right triangle with hypotenuse $r$. 6. This is a fundamental trigonometric relationship used to relate sides and angles in right triangles.