In the context of right triangles, which of the following best describes when the $\tan$ function can be applied to relate a line and an arc?

• A. The $\tan$ function can be used to relate the length of a line segment to the measure of an angle subtended by an arc in a right triangle.
• B. The $\tan$ function relates the area of an arc to the length of a line in any triangle.
• C. The $\tan$ function is used exclusively for calculating the circumference of an arc.
• D. The $\tan$ function is only applicable to circles and cannot be used with right triangles.

Answer: (A)

Verified step by step guidance

1. Recall that the tangent function in trigonometry is defined in a right triangle as the ratio of the length of the side opposite an angle to the length of the side adjacent to that angle. Mathematically, this is expressed as \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\).
2. Understand that in a right triangle, an angle can be associated with an arc measure (in degrees or radians) that subtends that angle at the vertex of the triangle.
3. Recognize that the tangent function connects a linear measurement (length of a side) to an angular measurement (the angle subtended by an arc), making it useful for relating a line segment length to the measure of an angle in a right triangle.
4. Note that the tangent function does not relate areas or circumferences directly; it specifically relates side lengths and angles in right triangles.
5. Therefore, the tangent function can be applied to relate the length of a line segment to the measure of an angle subtended by an arc in a right triangle, but it is not used exclusively for arcs or circles outside this context.