Question

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

Explanation:

Step1: Recall trigonometric - ratio definitions

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

Step2: Identify sides for $\tan$

For $\tan\theta = \frac{5}{12}$, the opposite side to the angle $\theta$ has length 5 and the adjacent side has length 12. In right - triangle $RST$, if we consider angle $T$, the opposite side to $T$ is $RS = 10$ and the adjacent side is $ST=24$. Reducing the ratio $\frac{RS}{ST}=\frac{10}{24}=\frac{5}{12}$. So the angle for $\tan$ is $\angle T$.

Step3: Identify sides for $\sin$

For $\sin\theta=\frac{12}{13}$, the opposite side to the angle $\theta$ has length 12 and the hypotenuse has length 13. In right - triangle $RST$, if we consider angle $R$, the opposite side to $R$ is $ST = 24$ and the hypotenuse is $RT = 26$. Reducing the ratio $\frac{ST}{RT}=\frac{24}{26}=\frac{12}{13}$. So the angle for $\sin$ is $\angle R$.

Answer:

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