In the context of right triangles, what does the trigonometric function $\tan$ (often abbreviated as tg) of an angle represent?

• A. The ratio of the length of the side opposite the angle to the length of the hypotenuse $\frac{\mathrm{opposite}}{\mathrm{hypotenuse}}$
• B. The ratio of the length of the hypotenuse to the length of the side opposite the angle $\frac{\mathrm{hypotenuse}}{\mathrm{opposite}}$
• C. The ratio of the length of the side opposite the angle to the length of the side adjacent to the angle $\frac{\mathrm{opposite}}{\mathrm{adjacent}}$
• D. The ratio of the length of the side adjacent to the angle to the length of the hypotenuse $\frac{\mathrm{adjacent}}{\mathrm{hypotenuse}}$

Answer: (C)

Verified step by step guidance

1. Recall that in a right triangle, the trigonometric functions sine, cosine, and tangent relate the angles to the ratios of the sides.
2. Identify the sides relative to the angle in question: the side opposite the angle, the side adjacent to the angle, and the hypotenuse (the longest side opposite the right angle).
3. Understand that the tangent function, denoted as \(\tan(\theta)\) or sometimes \(\tg(\theta)\), is defined as the ratio of the length of the side opposite the angle \(\theta\) to the length of the side adjacent to the angle \(\theta\).
4. Express this relationship mathematically as: \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\).
5. Note that this distinguishes tangent from sine and cosine, which involve the hypotenuse in their ratios.