Question

find ef.

triangle def with right angle at e, hypotenuse df = √86, angle at f is 42°

write your answer as an integer or as a decimal rounded to the nearest tenth.

ef =

submit

Explanation:

Step1: Identify triangle type and trigonometric ratio

We have a right - triangle \(DEF\) with \(\angle E = 90^{\circ}\), hypotenuse \(DF=\sqrt{86}\), and \(\angle F = 42^{\circ}\). We want to find the length of \(EF\), which is the adjacent side to \(\angle F\). The cosine of an angle in a right - triangle is defined as \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). So, \(\cos(42^{\circ})=\frac{EF}{DF}\).

Step2: Substitute the known values and solve for \(EF\)

We know that \(DF = \sqrt{86}\approx9.2736\) and \(\cos(42^{\circ})\approx0.7431\). Substituting into the formula \(\cos(42^{\circ})=\frac{EF}{\sqrt{86}}\), we get \(EF=\sqrt{86}\times\cos(42^{\circ})\).

First, calculate \(\sqrt{86}\approx9.2736\). Then, multiply by \(\cos(42^{\circ})\approx0.7431\): \(EF\approx9.2736\times0.7431\approx6.9\)

Answer:

\(6.9\)