Question

which trigonometric ratios are correct for triangle def? choose three correct answers.
cos(e) = 7/25
tan(d) = 24/7
sin(e) = 7/25
sin(d) = 24/25
tan(d) = 7/24

Explanation:

Step1: Find hypotenuse DE

In right triangle DEF, \( DF = 7 \), \( EF = 24 \). By Pythagorean theorem, \( DE=\sqrt{DF^{2}+EF^{2}}=\sqrt{7^{2}+24^{2}}=\sqrt{49 + 576}=\sqrt{625}=25 \).

Step2: Analyze \( \sin(D) \)

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

Step3: Analyze \( \sin(E) \)

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

Step4: Analyze \( \tan(D) \)

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

Step5: 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.

Step6: Analyze \( \tan(D)=\frac{7}{24} \)

From step4, \( \tan(D)=\frac{24}{7}
eq\frac{7}{24} \), so this is incorrect.

Answer:

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