Question

use the diagram and side lengths of triangle rst to determine the angles used for the trigonometric ratios.
$sin(square)=\frac{12}{13}$
$\tan(square)=\frac{5}{12}$

Explanation:

Step1: Simplify $\frac{12}{13}$ to match sides

First, scale the fraction to fit the triangle's sides: $\frac{12}{13} = \frac{24}{26}$. For sine, $\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}$. For angle $R$, opposite side is $24$, hypotenuse is $26$. So $\sin(R)=\frac{24}{26}=\frac{12}{13}$.

Step2: Simplify $\frac{5}{12}$ to match sides

Scale the fraction: $\frac{5}{12} = \frac{10}{24}$. For tangent, $\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$. For angle $T$, opposite side is $10$, adjacent side is $24$. So $\tan(T)=\frac{10}{24}=\frac{5}{12}$.

Answer:

$\sin(\boldsymbol{R}) = \frac{12}{13}$
$\tan(\boldsymbol{T}) = \frac{5}{12}$