Question

find ( mangle s ).

triangle rst with right angle at t, rs = 8.2, rt = 7.3

write your answer as an integer or as a decimal rounded to the nearest tenth.
( mangle s = square^circ )

Explanation:

Step1: Identify triangle type

Triangle \( RTS \) is right - angled at \( T \). We know the hypotenuse \( RS = 8.2 \) and the side \( RT=7.3 \). We want to find \( m\angle S \). In a right - triangle, we can use the sine function. The sine of an angle in a right - triangle is defined as \( \sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}} \). For \( \angle S \), the opposite side is \( RT = 7.3 \) and the hypotenuse is \( RS=8.2 \).

Step2: Apply sine formula

We use the formula \( \sin(\angle S)=\frac{RT}{RS} \). Substituting the values \( RT = 7.3 \) and \( RS = 8.2 \), we get \( \sin(\angle S)=\frac{7.3}{8.2}\approx0.8902 \).

Step3: Find the angle

To find \( \angle S \), we take the inverse sine (arcsin) of \( 0.8902 \). So \( \angle S=\arcsin(0.8902) \). Using a calculator, \( \arcsin(0.8902)\approx62.9^{\circ} \) (rounded to the nearest tenth).

Answer:

\( 62.9 \)