Question

find ( mangle f ).

triangle image with right angle at e, ed = 7.7, fd = 8.9

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

Explanation:

Step1: Identify triangle type and sides

We have a right - triangle \( \triangle DEF \) with \( \angle E = 90^{\circ} \), \( ED = 7.7 \) (adjacent to \( \angle F \)) and \( FD=8.9 \) (hypotenuse). We can use the cosine function to find \( \angle F \), since \( \cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}} \). Here, for \( \angle F \), the adjacent side is \( EF \)? Wait, no, wait. Wait, in right - triangle \( DEF \), right - angled at \( E \), the sides: \( ED = 7.7 \) (opposite to \( \angle F \))? Wait, no. Let's re - label: In right - triangle \( DEF \), right angle at \( E \), so:

• \( \angle E=90^{\circ} \)
• \( ED \): one leg (length \( 7.7 \))
• \( EF \): another leg
• \( FD \): hypotenuse (length \( 8.9 \))

We can use the sine function for \( \angle F \), because \( \sin(\angle F)=\frac{\text{opposite}}{\text{hypotenuse}} \). The side opposite to \( \angle F \) is \( ED = 7.7 \), and the hypotenuse is \( FD = 8.9 \). So \( \sin(\angle F)=\frac{ED}{FD}=\frac{7.7}{8.9} \)

Step2: Calculate the angle

First, calculate \( \frac{7.7}{8.9}\approx0.8652 \)

Then, to find \( \angle F \), we take the inverse sine (arcsin) of \( 0.8652 \). So \( \angle F=\arcsin(0.8652) \)

Using a calculator, \( \arcsin(0.8652)\approx59.9^{\circ} \) (rounded to the nearest tenth)

Answer:

\( 59.9 \)