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 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 \( AC \).
So, \( \tan A=\frac{\text{opposite}}{\text{adjacent}}=\frac{BC}{AC} \)

Step2: Identify the lengths of \( BC \) and \( AC \)

From the triangle, \( BC = 14 \) and \( AC=\sqrt{93} \)

Step3: Substitute the values into the tangent formula

\( \tan A=\frac{14}{\sqrt{93}} \)

Step4: Rationalize the denominator

To rationalize the denominator, multiply the numerator and denominator by \( \sqrt{93} \):
\( \tan A=\frac{14\times\sqrt{93}}{\sqrt{93}\times\sqrt{93}}=\frac{14\sqrt{93}}{93} \)

Answer:

\( \frac{14\sqrt{93}}{93} \)