Question

which trigonometric ratios are correct for triangle def? select three options.
□ sin(d) = \frac{24}{25}
□ cos(e) = \frac{7}{25}
□ tan(d) = \frac{24}{7}
□ sin(e) = \frac{7}{25}
□ tan(d) = \frac{7}{24}
(note: triangle def is a right triangle with right angle at f, df = 7, ef = 24, vertices d, f, e)

Explanation:

First, find the hypotenuse \( DE \) using the Pythagorean theorem. In right triangle \( DEF \), \( DF = 7 \), \( EF = 24 \), so \( DE=\sqrt{7^{2}+24^{2}}=\sqrt{49 + 576}=\sqrt{625}=25 \).

Step 1: Analyze \( \sin(D) \)

\( \sin(D)=\frac{\text{opposite to } D}{\text{hypotenuse}}=\frac{EF}{DE}=\frac{24}{25} \). So this is correct.

Step 2: Analyze \( \cos(E) \)

\( \cos(E)=\frac{\text{adjacent to } E}{\text{hypotenuse}}=\frac{EF}{DE}=\frac{24}{25}
eq\frac{7}{25} \). So this is incorrect.

Step 3: Analyze \( \tan(D) \)

\( \tan(D)=\frac{\text{opposite to } D}{\text{adjacent to } D}=\frac{EF}{DF}=\frac{24}{7} \). So this is correct.

Step 4: Analyze \( \sin(E) \)

\( \sin(E)=\frac{\text{opposite to } E}{\text{hypotenuse}}=\frac{DF}{DE}=\frac{7}{25} \). So this is correct.

Step 5: Analyze \( \tan(D)=\frac{7}{24} \)

From step 3, \( \tan(D)=\frac{24}{7}
eq\frac{7}{24} \). So this is incorrect.

Answer:

\( \sin(D)=\frac{24}{25} \), \( \tan(D)=\frac{24}{7} \), \( \sin(E)=\frac{7}{25} \) (corresponding to the first, third, and fourth options)