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

Answer:

$\frac{\sqrt{2}}{4}$

Explanation:

Step1: Recall tangent - ratio formula

$\tan A=\frac{\text{opposite}}{\text{adjacent}}$

Step2: Identify opposite and adjacent sides

In right - triangle $ABC$ with respect to angle $A$, the opposite side to angle $A$ is $BC = 2$ and the adjacent side to angle $A$ is $AC=\sqrt{32}$.

Step3: Simplify the tangent value

$\tan A=\frac{2}{\sqrt{32}}=\frac{2}{\sqrt{16\times2}}=\frac{2}{4\sqrt{2}}=\frac{1}{2\sqrt{2}}$. Rationalize the denominator: $\frac{1}{2\sqrt{2}}\times\frac{\sqrt{2}}{\sqrt{2}}=\frac{\sqrt{2}}{4}$.