find m∠s. write your answer as an integer or as a decimal rounded to the nearest tenth. m∠s = °

Explanation:

Step1: Identify the trigonometric ratio

In right - triangle $TUS$ with right - angle at $U$, we know the opposite side to $\angle S$ is $TU = 10$ and the hypotenuse is $TS$. First, find $TS$ using the Pythagorean theorem $TS=\sqrt{10^{2}+7^{2}}=\sqrt{100 + 49}=\sqrt{149}\approx12.2$. Then, we use the sine ratio $\sin S=\frac{opposite}{hypotenuse}=\frac{TU}{TS}$. Here, we can also use the inverse - sine function directly with the given sides. $\sin S=\frac{TU}{TS}$, and since we know $TU = 10$ and $TS$ (or we can use the tangent ratio if we consider the non - hypotenuse sides). Using the tangent ratio $\tan S=\frac{TU}{US}=\frac{10}{7}$.

Step2: Calculate the angle measure

We know that if $\tan S=\frac{10}{7}$, then $S=\arctan(\frac{10}{7})$. Using a calculator, $S=\arctan(\frac{10}{7})\approx55.0^{\circ}$.

Answer:

$55.0$