evaluate. write your answer in simplified, rationalized form. do not round.\n$\\tan\\left(\\frac{\\pi}{6}\\ri…

Explanation:

Step1: Recall the tangent value of special angles

We know that (\tan\left(\frac{\pi}{6}\right)=\frac{\sin\left(\frac{\pi}{6}\right)}{\cos\left(\frac{\pi}{6}\right)}). Since (\sin\left(\frac{\pi}{6}\right)=\frac{1}{2}) and (\cos\left(\frac{\pi}{6}\right)=\frac{\sqrt{3}}{2}).

Step2: Calculate the ratio

[ \begin{align*} \tan\left(\frac{\pi}{6}\right)&=\frac{\sin\left(\frac{\pi}{6}\right)}{\cos\left(\frac{\pi}{6}\right)}\ &=\frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}\ &=\frac{1}{\sqrt{3}} \end{align*} ]

Step3: Rationalize the denominator

Multiply the numerator and denominator by (\sqrt{3}): [ \begin{align*} \frac{1}{\sqrt{3}}&=\frac{1\times\sqrt{3}}{\sqrt{3}\times\sqrt{3}}\ &=\frac{\sqrt{3}}{3} \end{align*} ]

Answer:

(\frac{\sqrt{3}}{3})