Given a right triangle DEF with right angle at $E$, which of the following trigonometric ratios are correct for angle $D$? Select the correct option.

• A. $\sin D=\frac{\mathrm{adjacent}}{\mathrm{opposite}}$
• B. $\tan D=\frac{\mathrm{opposite}}{\mathrm{adjacent}}$
• C. $\sin D=\frac{\mathrm{opposite}}{\mathrm{hypotenuse}}$
• D. $\cos D=\frac{\mathrm{adjacent}}{\mathrm{hypotenuse}}$

Answer: (D)

Verified step by step guidance

1. Identify the sides of the right triangle DEF relative to angle D. The side opposite angle D is the side across from D, the adjacent side is the one next to D (but not the hypotenuse), and the hypotenuse is the longest side opposite the right angle at E.
2. Recall the definitions of the primary trigonometric ratios for an angle \( \theta \) in a right triangle: \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \), \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \), and \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).
3. Examine each given ratio and compare it to the standard definitions. For example, if the problem states \( \sin(D) = \frac{\text{adjacent}}{\text{opposite}} \), check if this matches the definition of sine, which it does not.
4. Confirm that the correct expressions for angle D are those that match the standard definitions: \( \sin(D) = \frac{\text{opposite}}{\text{hypotenuse}} \), \( \cos(D) = \frac{\text{adjacent}}{\text{hypotenuse}} \), and \( \tan(D) = \frac{\text{opposite}}{\text{adjacent}} \).
5. Select the correct trigonometric ratios for angle D based on these definitions, ensuring that the ratio's numerator and denominator correspond to the correct sides relative to angle D.