SAT Question 053e5e9f: Right Triangles And Trigonometry answer and explanation with Desmos solution

Choice D is correct. The figure shows that triangle $QRS$ is a right triangle. Each of the given choices is an expression containing $\sin Q$ or $\cos Q$. For an acute angle in a right triangle, the sine of the angle is the length of the opposite leg divided by the length of the hypotenuse, and the cosine of the angle is the length of the adjacent leg divided by the length of the hypotenuse. It follows that $\sin Q=\frac{RS}{QS}$ and $\cos Q=\frac{QR}{QS}$, where $QR<RS$. It's given in the figure that the length of side $RS$ is $25$. Since only the equation $\sin Q=\frac{RS}{QS}$ contains $RS$, an expression representing $QS$ can be found by substituting $25$ for $RS$ in this equation, which yields $\sin Q=\frac{25}{QS}$. Multiplying both sides of this equation by $QS$ yields $QS\cdot \sin Q=25$. Dividing both sides of this equation by $\sin Q$ yields $QS=\frac{25}{\sin Q}$. Therefore, the expression $\frac{25}{\sin Q}$ represents the length of $PARS{E}_{E}RROR:Entit{y}^{\prime }mac{r}^{\prime }notdefined,line1,column139(<string>,line1)$.