Question

find m∠e.
write your answer as an integer or as a decimal rounded to the nearest tenth.
m∠e = °

Explanation:

Step1: Identify triangle type and sides

Triangle \( FDE \) is right - angled at \( D \). Hypotenuse \( FE = 10 \), opposite side to \( \angle E \) is \( FD \), adjacent side to \( \angle E \) is \( DE = 4 \). First, find \( FD \) using Pythagorean theorem: \( FD=\sqrt{FE^{2}-DE^{2}}=\sqrt{10^{2}-4^{2}}=\sqrt{100 - 16}=\sqrt{84}\approx9.165 \). But we can also use trigonometric ratios. For \( \angle E \), \( \cos\angle E=\frac{DE}{FE}=\frac{4}{10} = 0.4 \) or \( \sin\angle E=\frac{FD}{FE}\), but using cosine is easier here.

Step2: Calculate \( \angle E \)

We know that \( \cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}} \). For \( \angle E \), adjacent side \( DE = 4 \), hypotenuse \( FE = 10 \). So \( \cos\angle E=\frac{4}{10}=0.4 \). Then \( \angle E=\cos^{- 1}(0.4) \). Using a calculator, \( \cos^{-1}(0.4)\approx66.4^{\circ} \) (or we can use sine: \( \sin\angle E=\frac{FD}{FE}=\frac{\sqrt{84}}{10}\approx\frac{9.165}{10} = 0.9165 \), \( \angle E=\sin^{-1}(0.9165)\approx66.4^{\circ} \))

Answer:

\( 66.4 \)