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}\\)
(image of right triangle def with right angle at f, df = 7, ef = 24)

Explanation:

Step1: Find hypotenuse DE

Use Pythagorean theorem:
$$DE = \sqrt{DF^2 + EF^2} = \sqrt{7^2 + 24^2} = \sqrt{49+576} = \sqrt{625} = 25$$

Step2: Evaluate sin(D)

$\sin(D)=\frac{\text{opposite to }D}{\text{hypotenuse}} = \frac{24}{25}$

Step3: Evaluate cos(E)

$\cos(E)=\frac{\text{adjacent to }E}{\text{hypotenuse}} = \frac{24}{25}$

Step4: Evaluate tan(D)

$\tan(D)=\frac{\text{opposite to }D}{\text{adjacent to }D} = \frac{24}{7}$

Step5: Evaluate sin(E)

$\sin(E)=\frac{\text{opposite to }E}{\text{hypotenuse}} = \frac{7}{25}$

Step6: Evaluate tan(D) (second option)

$\tan(D)=\frac{7}{24}$ is incorrect, as shown in Step4.

Answer:

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