find tv. write your answer as an integer or as a decimal rounded to the nearest tenth. tv =

Explanation:

Step1: Identify the trigonometric relation

In right - triangle $TUV$ with $\angle U = 90^{\circ}$, we know $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. Here, $\theta = 51^{\circ}$ and the side opposite to $\angle T$ is $UV=\sqrt{58}$, and we want to find the hypotenuse $TV$. So, $\sin T=\frac{UV}{TV}$.

Step2: Substitute the values

Since $\sin(51^{\circ})\approx0.777$ and $UV = \sqrt{58}\approx7.62$, and $\sin T=\frac{UV}{TV}$, we can rewrite it as $TV=\frac{UV}{\sin T}$. Substituting the values, we get $TV=\frac{\sqrt{58}}{\sin(51^{\circ})}$.

Step3: Calculate the value of TV

$TV=\frac{7.62}{0.777}\approx9.8$.

Answer:

$9.8$