Question

solve the equation. give a general formula for all the solutions. list all the solutions for ( k = 0,1,2,3,4 ), and 5
( \tan \theta=\frac{sqrt{3}}{3} )
write the general formula for all the solutions to ( \tan \theta=\frac{sqrt{3}}{3} ).
( \theta=square )
(simplify your answer. use angle measures greater than or equal to 0 and less than ( 2 pi ). type an exact answer, using ( pi ) as nee
expression using ( k ) as the variable.)

Explanation:

Step1: Find the reference angle

We know that \(\tan\theta=\frac{\sqrt{3}}{3}\), and \(\tan\frac{\pi}{6}=\frac{\sqrt{3}}{3}\). The tangent function \(y = \tan x\) has a period of \(\pi\).

Step2: Write the general formula

The general formula for the solutions of the equation \(\tan\theta = \tan\alpha\) is \(\theta=\alpha + k\pi\), where \(k\in\mathbb{Z}\). Since \(\alpha=\frac{\pi}{6}\), the general formula for the solutions of \(\tan\theta=\frac{\sqrt{3}}{3}\) is \(\theta=\frac{\pi}{6}+k\pi\), where \(k = 0,\pm1,\pm2,\cdots\)

Answer:

\(\theta=\frac{\pi}{6}+k\pi\)