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}\\)

Explanation:

First, we need to find the hypotenuse of triangle DEF using the Pythagorean theorem. In a right triangle, \( c = \sqrt{a^2 + b^2} \), where \( a = 7 \) and \( b = 24 \).

Step 1: Calculate the hypotenuse \( DE \)

\( DE = \sqrt{7^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25 \)

Step 2: Analyze \( \sin(D) \)

For angle \( D \), the opposite side is \( EF = 24 \), and the hypotenuse is \( DE = 25 \). So, \( \sin(D) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{24}{25} \). This is correct.

Step 3: Analyze \( \cos(E) \)

For angle \( E \), the adjacent side is \( EF = 24 \), and the hypotenuse is \( DE = 25 \). So, \( \cos(E) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{24}{25} \), not \( \frac{7}{25} \). This is incorrect.

Step 4: Analyze \( \tan(D) \)

For angle \( D \), the opposite side is \( EF = 24 \), and the adjacent side is \( DF = 7 \). So, \( \tan(D) = \frac{\text{opposite}}{\text{adjacent}} = \frac{24}{7} \). This is correct.

Step 5: Analyze \( \sin(E) \)

For angle \( E \), the opposite side is \( DF = 7 \), and the hypotenuse is \( DE = 25 \). So, \( \sin(E) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{7}{25} \). This is correct.

Step 6: Analyze \( \tan(D) = \frac{7}{24} \)

This is the reciprocal of the correct tangent for angle \( D \), so it's incorrect.

Answer:

The correct options are:

• \( \sin(D) = \frac{24}{25} \)
• \( \tan(D) = \frac{24}{7} \)
• \( \sin(E) = \frac{7}{25} \)