Question

use the diagram and side lengths of triangle rst to determine the angles used for the trigonometric ratios. sin( ) = 12/13 tan( r s t ) = 5/12

Explanation:

Step1: Recall sine ratio formula

$\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$
In right - triangle $RST$ with right - angle at $S$, the hypotenuse $RT = 26$.

Step2: Analyze the given sine value

We know that $\sin(\text{angle})=\frac{12}{13}=\frac{24}{26}$. The side opposite the angle and the hypotenuse are in the ratio $24:26$. The side opposite angle $T$ is $RS = 24$ and the hypotenuse is $RT=26$. So, $\sin(T)=\frac{RS}{RT}=\frac{24}{26}=\frac{12}{13}$.

Step3: Recall tangent ratio formula

$\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$
For $\tan(R)$, the side opposite $R$ is $ST = 24$ and the side adjacent to $R$ is $RS = 10$. So, $\tan(R)=\frac{ST}{RS}=\frac{24}{10}=\frac{12}{5}$. For $\tan(S)$ is undefined as $S = 90^{\circ}$. For $\tan(T)=\frac{RS}{ST}=\frac{10}{24}=\frac{5}{12}$.

Answer:

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