Question

recognizing trigonometric ratios
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: Recall sine definition

For an acute angle in a right triangle, $\sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}}$.
We know $\sin(\theta)=\frac{12}{13}=\frac{24}{26}$. The hypotenuse is $RT=26$, and the side opposite angle $R$ is $ST=24$. So $\sin(R)=\frac{24}{26}=\frac{12}{13}$.

Step2: Recall tangent definition

For an acute angle in a right triangle, $\tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}}$.
We know $\tan(\theta)=\frac{5}{12}=\frac{10}{24}$. For angle $T$, the opposite side is $RS=10$, and the adjacent side is $ST=24$. So $\tan(T)=\frac{10}{24}=\frac{5}{12}$.

Answer:

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