solve the equation. give a general formula for all the solutions. list all the solutions for ( k = 0,1,2,3,4…

solve the equation. give a general formula for all the solutions. list all the solutions for ( k = 0,1,2,3,4 ), and 5\n( \tan \theta=\frac{sqrt{3}}{3} )\nwrite the general formula for all the solutions to ( \tan \theta=\frac{sqrt{3}}{3} ).\n( \theta=square )\n(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\n expression using ( k ) as the variable.)

solve the equation. give a general formula for all the solutions. list all the solutions for ( k = 0,1,2,3,4 ), and 5\n( \tan \theta=\frac{sqrt{3}}{3} )\nwrite the general formula for all the solutions to ( \tan \theta=\frac{sqrt{3}}{3} ).\n( \theta=square )\n(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\n expression using ( k ) as the variable.)

Answer

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)