Question

find ef.
image of right triangle def with right angle at e, leg de labeled \\(\sqrt{70}\\), angle at d labeled 29°
write your answer as an integer or as a decimal rounded to the nearest tenth.
ef = \\(\square\\)

Explanation:

Step1: Identify the trigonometric ratio

In right triangle \( DEF \) (right - angled at \( E \)), we know the length of the adjacent side to the angle \( \angle D = 29^{\circ} \) (which is \( DE=\sqrt{70} \)) and we want to find the length of the opposite side \( EF \) with respect to \( \angle D \). The trigonometric ratio that relates the opposite side and the adjacent side of an angle in a right triangle is the tangent function. The formula for the tangent of an angle \( \theta \) in a right triangle is \( \tan\theta=\frac{\text{opposite}}{\text{adjacent}} \).

Step2: Apply the tangent formula

For \( \angle D = 29^{\circ} \), \( \text{opposite}=EF \) and \( \text{adjacent} = DE=\sqrt{70} \). So we have \( \tan(29^{\circ})=\frac{EF}{\sqrt{70}} \). To find \( EF \), we can rearrange the formula: \( EF=\sqrt{70}\times\tan(29^{\circ}) \).

First, calculate the value of \( \sqrt{70}\approx8.3666 \) and \( \tan(29^{\circ})\approx0.5543 \). Then multiply these two values: \( EF\approx8.3666\times0.5543\approx4.6 \) (rounded to the nearest tenth).

Answer:

\( 4.6 \)