Question

find the exact value of \\(\tan k\\) 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 defined as the ratio of the length of the opposite side to the length of the adjacent side. For angle \( K \) in right triangle \( KLM \) (with right angle at \( L \)):
\[
\tan K=\frac{\text{opposite side to } K}{\text{adjacent side to } K}
\]

Step2: Identify the opposite and adjacent sides to angle \( K \)

• The side opposite to angle \( K \) is \( ML \), which has length \( 7 \).
• The side adjacent to angle \( K \) is \( LK \), which has length \( \sqrt{51} \).

Step3: Calculate \( \tan K \)

Using the definition of tangent:
\[
\tan K = \frac{ML}{LK}=\frac{7}{\sqrt{51}}
\]
To rationalize the denominator, multiply the numerator and denominator by \( \sqrt{51} \):
\[
\tan K=\frac{7\times\sqrt{51}}{\sqrt{51}\times\sqrt{51}}=\frac{7\sqrt{51}}{51}
\]

Answer:

\(\dfrac{7\sqrt{51}}{51}\)