Questions: Find m angle E. Write your answer as an integer or as a decimal rounded to the nearest tenth. m angle E =

Solution Steps

We are given a triangle $\triangle DEF$ with a right angle at $D$. We are given the side lengths $DF = 4$ and $EF = 6$. We are asked to find $m\angle E$.

Since we have a right triangle, we can use trigonometric ratios. We are given the side adjacent to $\angle E$ and the hypotenuse. Therefore, we can use the cosine function.

$\cos(E) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{EF}{DF} = \frac{6}{4} = \frac{3}{2} = 1.5$

However, the cosine function cannot be greater than 1. Therefore, there must be an error in the problem statement. The side $EF$ should be $6$, but $DF$ cannot be $4$. It must be that $DF < EF$. Let's assume we are looking for $\angle F$. $\cos F = \frac{DF}{EF} = \frac{4}{6} = \frac{2}{3}$ $m\angle F = \arccos(\frac{2}{3}) \approx 48.2^\circ$

If we are looking for $\angle E$: $\cos E = \frac{ED}{EF}$ Since $\angle D = 90^\circ$, by the Pythagorean theorem: $ED^2 + DF^2 = EF^2$ $ED^2 + 4^2 = 6^2$ $ED^2 + 16 = 36$ $ED^2 = 20$ $ED = \sqrt{20} = 2\sqrt{5}$ $\cos E = \frac{2\sqrt{5}}{6} = \frac{\sqrt{5}}{3}$ $m\angle E = \arccos(\frac{\sqrt{5}}{3}) \approx 41.8^\circ$

Final Answer