Question

find ( mangle f ).
triangle def with right angle at e. de = 3.8, ef = 7.2. write your answer as an integer or as a decimal rounded to the nearest tenth. ( mangle f = square^circ )

Explanation:

Step1: Identify the trigonometric ratio

In right triangle \( DEF \) (right - angled at \( E \)), we know the lengths of the opposite side (\( DE = 3.8 \)) and the adjacent side (\( EF=7.2 \)) with respect to angle \( F \). The tangent of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the adjacent side. So, \( \tan(F)=\frac{DE}{EF} \).

Step2: Substitute the values

Substitute \( DE = 3.8 \) and \( EF = 7.2 \) into the tangent formula: \( \tan(F)=\frac{3.8}{7.2}\approx0.5278 \).

Step3: Find the angle

To find the measure of angle \( F \), we take the arctangent (inverse tangent) of \( 0.5278 \). Using a calculator, \( F=\arctan(0.5278) \). Calculating this, we get \( F\approx27.8^{\circ} \) (rounded to the nearest tenth).

Answer:

\( 27.8 \)