Which of the following statements is true about the tangent function $\tan$ in a right triangle?

• A. The tangent of an angle is the ratio of the adjacent side to the opposite side: $\tan \left(\theta \right)=\frac{\mathrm{adjacent}}{\mathrm{opposite}}$
• B. The tangent of an angle is the ratio of the hypotenuse to the adjacent side: $\tan \left(\theta \right)=\frac{\mathrm{hypotenuse}}{\mathrm{adjacent}}$
• C. The tangent of an angle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle: $\tan \left(\theta \right)=\frac{\mathrm{opposite}}{\mathrm{adjacent}}$
• D. The tangent of an angle is the ratio of the hypotenuse to the opposite side: $\tan \left(\theta \right)=\frac{\mathrm{hypotenuse}}{\mathrm{opposite}}$

Answer: (C)

Verified step by step guidance

1. Recall the definition of the tangent function in a right triangle. Tangent of an angle \( \theta \) is defined as the ratio of the length of the side opposite to \( \theta \) to the length of the side adjacent to \( \theta \).
2. Express this relationship mathematically as: \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \)
3. Understand that the hypotenuse is the longest side in a right triangle and is not used in the tangent ratio. Instead, it is used in sine and cosine ratios.
4. Review the incorrect options: - \( \tan(\theta) = \frac{\text{adjacent}}{\text{opposite}} \) reverses the ratio and is incorrect. - \( \tan(\theta) = \frac{\text{hypotenuse}}{\text{adjacent}} \) and \( \tan(\theta) = \frac{\text{hypotenuse}}{\text{opposite}} \) involve the hypotenuse, which is not part of the tangent definition.
5. Conclude that the correct statement is: \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \)