Question

find the exact value of tan a in simplest radical form.

Explanation:

Step1: Recall the definition of tangent in a right triangle

In a right triangle, the tangent of an acute angle is the ratio of the length of the opposite side to the length of the adjacent side. For angle \( A \), the opposite side is \( BC \) and the adjacent side is \( AC \).

Step2: Identify the lengths of the opposite and adjacent sides

From the diagram, the length of the side opposite to angle \( A \) ( \( BC \)) is \( 4 \), and the length of the side adjacent to angle \( A \) ( \( AC \)) is \( 3 \).

Step3: Calculate \( \tan A \)

Using the definition of tangent, \( \tan A=\frac{\text{opposite}}{\text{adjacent}}=\frac{BC}{AC} \). Substituting the values, we get \( \tan A = \frac{4}{3} \).

Answer:

\( \frac{4}{3} \)