Question

consider right triangle △def below. which expressions represent the length of side \\(\overline{de}\\)?

Explanation:

Step1: Identify triangle parts

In right triangle \( \triangle DEF \), \( \angle D = 90^\circ \), \( \angle F = 40^\circ \), hypotenuse \( EF = 5 \), adjacent side to \( \angle F \) is \( DF = 3.8 \), opposite side to \( \angle F \) is \( DE \).

Step2: Use trigonometric ratios

• Sine: \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \), so \( \sin(40^\circ) = \frac{DE}{5} \Rightarrow DE = 5\sin(40^\circ) \).
• Cosine: \( \cos(50^\circ) = \frac{\text{opposite to } 50^\circ}{\text{hypotenuse}} \) (since \( \angle E = 50^\circ \)), so \( \cos(50^\circ) = \frac{DE}{5} \Rightarrow DE = 5\cos(50^\circ) \).
• Tangent: \( \tan(40^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{DE}{3.8} \Rightarrow DE = 3.8\tan(40^\circ) \).

Answer:

Possible expressions: \( 5\sin(40^\circ) \), \( 5\cos(50^\circ) \), \( 3.8\tan(40^\circ) \) (any of these, depending on context; e.g., \( 5\sin(40^\circ) \) is a common one).