Which equation correctly relates the measure of angle ${\theta }_1$ in a right triangle to the lengths of the opposite side $O$ and the hypotenuse $H$?

• A. $tan\left({\theta }_1\right)=\frac{H}{O}$
• B. $cos\left({\theta }_1\right)=\frac{O}{H}$
• C. $sin\left({\theta }_1\right)=\frac{O}{H}$
• D.

Answer: (C)

Verified step by step guidance

1. Recall the definitions of the primary trigonometric ratios in a right triangle: sine, cosine, and tangent. These relate an angle to ratios of specific sides.
2. Identify the sides relative to the angle \( \theta_1 \): the opposite side (O) is the side opposite to \( \theta_1 \), the adjacent side is next to \( \theta_1 \), and the hypotenuse (H) is the longest side opposite the right angle.
3. The sine of an angle is defined as the ratio of the length of the opposite side to the hypotenuse, expressed as \( \sin(\theta_1) = \frac{O}{H} \).
4. The cosine of an angle is the ratio of the adjacent side to the hypotenuse, \( \cos(\theta_1) = \frac{\text{Adjacent}}{H} \), and the tangent is the ratio of the opposite side to the adjacent side, \( \tan(\theta_1) = \frac{O}{\text{Adjacent}} \).
5. Compare the given options to these definitions to determine which equation correctly relates \( \theta_1 \) to the sides O and H. The correct equation matches the sine definition: \( \sin(\theta_1) = \frac{O}{H} \).