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:

Step1: Find the hypotenuse

Using Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(a = 7\), \(b=24\). Then \(c=\sqrt{7^{2}+24^{2}}=\sqrt{49 + 576}=\sqrt{625}=25\)

Step2: Recall trigonometric ratios

• \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\), \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\)
• For \(\angle D\):
• Opposite side \(= 24\), adjacent side \(=7\), hypotenuse \(=25\)
• \(\sin(D)=\frac{24}{25}\), \(\tan(D)=\frac{24}{7}\)
• For \(\angle E\):
• Opposite side \(=7\), adjacent side \(=24\), hypotenuse \(=25\)
• \(\cos(E)=\frac{7}{25}\), \(\sin(E)=\frac{7}{25}\)

Answer:

\(\sin(D)=\frac{24}{25}\), \(\cos(E)=\frac{7}{25}\), \(\tan(D)=\frac{24}{7}\), \(\sin(E)=\frac{7}{25}\) are the relevant ratios. So the correct options are \(\sin(D)=\frac{24}{25}\), \(\cos(E)=\frac{7}{25}\), \(\tan(D)=\frac{24}{7}\), \(\sin(E)=\frac{7}{25}\) (assuming the options are labeled as in the problem statement, but since no labels are given in the problem's text - based on calculation these are the correct ratio expressions)