In a right triangle, what is the definition of $\sin$ of an angle $\theta$?

• A. The ratio of the length of the hypotenuse to the length of the side opposite angle $\theta$
• B. The ratio of the length of the side adjacent to angle $\theta$ to the length of the hypotenuse
• C. The ratio of the length of the side opposite angle $\theta$ to the length of the side adjacent to angle $\theta$
• D. The ratio of the length of the side opposite angle $\theta$ to the length of the hypotenuse

Answer: (D)

Verified step by step guidance

1. Recall that in a right triangle, the sine of an angle \( \theta \) is defined as the ratio of the length of the side opposite to \( \theta \) to the length of the hypotenuse.
2. Identify the side opposite to the angle \( \theta \) in the triangle. This is the side that does not form the angle \( \theta \) but is directly across from it.
3. Identify the hypotenuse, which is the longest side of the right triangle and is always opposite the right angle.
4. Write the sine ratio as \( \sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}} \).
5. Understand that this ratio helps relate the angle \( \theta \) to the lengths of the sides in the triangle, which is fundamental in solving many trigonometric problems.