Step 1: Problem
Find E and find F to the nearest degree and find DF to the nearest tenth.
Step 2: Concept
1. Apply sine and cosine formula to find the angle E and F
[tex]\begin{gathered} \sin \theta\text{ = }\frac{opposite}{\text{hypotenuse}} \\ \cos \theta\text{ = }\frac{adjacent}{\text{hypotenuse}} \end{gathered}[/tex]
2. Apply Pythagoras theorem to find length DF
[tex]\text{Opposite}^2+Adjacent^2=Hypotenuse^2[/tex]
Step 3:
For angle F
Opposite = 5
Hypotenuse = 5.83
[tex]\begin{gathered} \sin F\text{ = }\frac{Opposite}{\text{Hypotenuse}} \\ \sin F\text{ = }\frac{5}{5.83} \\ \sin F\text{ = 0.857632} \\ F\text{ = }\sin ^{-1}0.857632 \\ F\text{ = 59.05} \\ F\text{ = 59} \end{gathered}[/tex]
For angle E
adjacent = 5
hypotenuse = 5.83
[tex]\begin{gathered} \cos E\text{ = }\frac{adjacent}{\text{hypotenuse}} \\ \cos E\text{ = }\frac{5}{8.83} \\ \cos E\text{ = 0.857632} \\ E\text{ = }\cos ^{-1}0.857632 \\ E\text{ = 30.9} \\ E\text{ = 31} \end{gathered}[/tex]
Hypotenuse = 5.85 ft
Opposite = 5ft
Adjacent = DF
[tex]\begin{gathered} \text{Opposite}^2+Adjacent^2=Hypotenuse^2 \\ 5^2+DF^2=5.83^2 \\ 25+DF^2\text{ = 33.9889} \\ DF^2\text{ = 33.9889 - 25} \\ DF^2\text{ = 8.9889} \\ DF\text{ = }\sqrt[]{8.9889} \\ DF\text{ = 2.998} \\ DF\text{ = 3.0} \end{gathered}[/tex]
Step 4: Final answer
Angle F = 59
Angle E = 31
Length DF = 3.0
