Question

which ratio has a value equal to $\frac{12}{13}$?
$sin(r)$
$sin(p)$
$\tan(p)$
$cos(r)$

Explanation:

Step1: Recall trigonometric - ratio definitions

In a right - triangle, $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$, and $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$.

Step2: Analyze for angle $R$

For $\angle R$, the opposite side is $PQ = 12$ and the hypotenuse is $PR=13$. So, $\sin(R)=\frac{PQ}{PR}=\frac{12}{13}$.

Step3: Analyze for angle $P$

For $\angle P$, the opposite side is $RQ = 5$ and the hypotenuse is $PR = 13$, so $\sin(P)=\frac{RQ}{PR}=\frac{5}{13}$. Also, $\tan(P)=\frac{RQ}{PQ}=\frac{5}{12}$.

Step4: Analyze $\cos(R)$

For $\angle R$, the adjacent side is $RQ = 5$ and the hypotenuse is $PR = 13$, so $\cos(R)=\frac{RQ}{PR}=\frac{5}{13}$.

Answer:

$\sin(R)$