Question

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

Explanation:

Step1: Identify the triangle type and sides

We have a right - triangle \( \triangle FGH \) with \( \angle G = 90^{\circ} \), hypotenuse \( FH = 8 \) and opposite side to \( \angle F \) is \( GH = 5.6 \). We can use the sine function to find \( \angle F \). The sine of an angle \( \theta \) in a right - triangle is defined as \( \sin\theta=\frac{\text{opposite}}{\text{hypotenuse}} \). So for \( \angle F \), \( \sin(\angle F)=\frac{GH}{FH} \)

Step2: Substitute the values

Substitute \( GH = 5.6 \) and \( FH = 8 \) into the sine formula: \( \sin(\angle F)=\frac{5.6}{8}=0.7 \)

Step3: Find the angle

To find \( \angle F \), we take the inverse sine (arcsin) of \( 0.7 \). So \( \angle F=\arcsin(0.7) \)
Using a calculator, \( \arcsin(0.7)\approx44.4^{\circ} \) (rounded to the nearest tenth)

Answer:

\( 44.4 \)