Question

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

Explanation:

Step1: Recall the definition of tangent in a right - triangle

In a right - triangle, \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). For \(\angle L\), the side opposite to \(\angle L\) is \(MN\) and the side adjacent to \(\angle L\) is \(ML\).

Step2: Identify the lengths of the opposite and adjacent sides

We are given \(MN = \sqrt{52}\) and \(ML=12\).

Step3: Simplify \(\sqrt{52}\)

We know that \(52 = 4\times13\), so \(\sqrt{52}=\sqrt{4\times13}=2\sqrt{13}\).

Step4: Calculate \(\tan L\)

\(\tan L=\frac{MN}{ML}=\frac{\sqrt{52}}{12}=\frac{2\sqrt{13}}{12}\)

Step5: Reduce the fraction

Cancel out the common factor of 2 in the numerator and denominator. \(\frac{2\sqrt{13}}{12}=\frac{\sqrt{13}}{6}\)

Answer:

\(\frac{\sqrt{13}}{6}\)