Question

find the cosine of ∠e.
f 12 d
5
e
simplify your answer and write it as a proper fraction, improper fraction, or whole number.
cos (e) =
submit

Explanation:

Step1: Identify triangle type and sides

This is a right - triangle \( \triangle EFD \) with \( \angle F = 90^{\circ} \). The side adjacent to \( \angle E \) is \( EF = 5 \), the side opposite to \( \angle E \) is \( FD=12 \), and we need to find the hypotenuse \( ED \) first using the Pythagorean theorem. The Pythagorean theorem states that for a right - triangle \( a^{2}+b^{2}=c^{2} \), where \( c \) is the hypotenuse and \( a,b \) are the legs. So, \( ED^{2}=EF^{2}+FD^{2} \). Substituting \( EF = 5 \) and \( FD = 12 \), we get \( ED^{2}=5^{2}+12^{2}=25 + 144=169 \), so \( ED=\sqrt{169} = 13 \).

Step2: Recall the definition of cosine

The cosine of an angle in a right - triangle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse. For \( \angle E \), the adjacent side is \( EF = 5 \) and the hypotenuse is \( ED=13 \). So, \( \cos(E)=\frac{\text{Adjacent}}{\text{Hypotenuse}}=\frac{EF}{ED} \).

Step3: Calculate \( \cos(E) \)

Substituting \( EF = 5 \) and \( ED = 13 \) into the formula for cosine, we get \( \cos(E)=\frac{5}{13} \).

Answer:

\( \frac{5}{13} \)