Question

find ef.
triangle image with right angle at e, angle at f is 25°, ed is √7
write your answer as an integer or as a decimal rounded to the nearest tenth.
ef =
submit
evaluate. write your answer in simplified, rationalized form. do not round.
cos 30° =

Explanation:

Step1: Identify the trigonometric ratio

In right triangle \( DEF \), \( \angle E = 90^\circ \), \( \angle F = 25^\circ \), and the side opposite to \( \angle F \) is \( DE=\sqrt{7} \), and \( EF \) is the adjacent side to \( \angle F \). We use the tangent function? Wait, no, wait. Wait, \( \sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}} \), \( \cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}} \), \( \tan(\theta)=\frac{\text{opposite}}{\text{adjacent}} \). Wait, no, in triangle \( DEF \), angle at \( F \) is \( 25^\circ \), right angle at \( E \), so \( DE \) is opposite to \( \angle F \), \( EF \) is adjacent to \( \angle F \), and \( DF \) is hypotenuse. Wait, we need to find \( EF \). Wait, \( \tan(25^\circ)=\frac{DE}{EF} \)? Wait, no: \( \tan(\theta)=\frac{\text{opposite}}{\text{adjacent}} \), so \( \tan(25^\circ)=\frac{DE}{EF} \), so \( EF = \frac{DE}{\tan(25^\circ)} \). Wait, \( DE = \sqrt{7} \approx 2.6458 \). \( \tan(25^\circ) \approx 0.4663 \). So \( EF=\frac{\sqrt{7}}{\tan(25^\circ)} \approx \frac{2.6458}{0.4663} \approx 5.7 \)? Wait, no, wait, maybe I mixed up. Wait, no, let's check again. Wait, angle at \( F \) is \( 25^\circ \), so the side opposite to \( 25^\circ \) is \( DE \) (length \( \sqrt{7} \)), and the side adjacent to \( 25^\circ \) is \( EF \). So \( \tan(25^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{DE}{EF} \), so \( EF = \frac{DE}{\tan(25^\circ)} \). Let's compute that. First, \( \sqrt{7} \approx 2.6458 \), \( \tan(25^\circ) \approx 0.4663 \). So \( 2.6458 / 0.4663 \approx 5.7 \). Wait, but let's do it more accurately. Alternatively, maybe using cosine? Wait, no, hypotenuse is \( DF \), but we don't know that. So tangent is the way. So \( EF = \frac{\sqrt{7}}{\tan(25^\circ)} \). Let's calculate:

\( \sqrt{7} \approx 2.645751311 \)

\( \tan(25^\circ) \approx 0.46630765815 \)

So \( EF \approx \frac{2.645751311}{0.46630765815} \approx 5.67 \), which rounds to 5.7. Wait, but let's check with calculator. Let's compute \( \sqrt{7} \approx 2.6458 \), \( \tan(25) \approx 0.4663 \), so 2.6458 / 0.4663 ≈ 5.67, which is 5.7 when rounded to nearest tenth.

Step2: Compute the value

So \( EF = \frac{\sqrt{7}}{\tan(25^\circ)} \approx \frac{2.6458}{0.4663} \approx 5.7 \)

Answer:

\( 5.7 \)