Given a right triangle DEF with right angle at F, if angle E is one of the acute angles, what is the value of $\sin \left(E\right)$ in terms of the sides of the triangle?

• A. $\frac{\mathrm{EF}}{\mathrm{DE}}$
• B. $\frac{\mathrm{DF}}{\mathrm{DE}}$
• C. $\frac{\mathrm{DE}}{\mathrm{DF}}$
• D. $\frac{\mathrm{DE}}{\mathrm{EF}}$

Answer: (B)

Verified step by step guidance

1. Identify the sides of the right triangle DEF with the right angle at F. The hypotenuse is the side opposite the right angle, which is DE.
2. Recall the definition of sine for an acute angle in a right triangle: \(\sin(\text{angle}) = \frac{\text{opposite side}}{\text{hypotenuse}}\).
3. Since angle E is one of the acute angles, determine which side is opposite to angle E. The side opposite angle E is DF.
4. Using the sine definition, express \(\sin(E)\) as the ratio of the length of side DF (opposite to E) over the hypotenuse DE.
5. Write the formula: \(\sin(E) = \frac{DF}{DE}\), which expresses sine of angle E in terms of the sides of the triangle.