Question

find all solutions to the equation.

\\( \tan \left( \frac { 1 } { 2 } x \
ight) = 0 \\)

write your answer in radians in terms of \\( \pi \\), and use the \or\ button as necessary.

example: \\( x = \frac { \pi } { 5 } + 2 k \pi , k \in \mathbb { z } \\) or \\( x = \frac { \pi } { 7 } + k \pi , k \in \mathbb { z } \\)

\\( x = \square \\)

Explanation:

Step1: Recall the general solution of \(\tan\theta = 0\)

The general solution of \(\tan\theta=0\) is \(\theta = k\pi\), where \(k\in\mathbb{Z}\). In our equation, \(\theta=\frac{1}{2}x\). So we have \(\frac{1}{2}x = k\pi\).

Step2: Solve for \(x\)

Multiply both sides of the equation \(\frac{1}{2}x = k\pi\) by \(2\). Using the property of equality \(a = b\Rightarrow 2a=2b\), we get \(x = 2k\pi\), \(k\in\mathbb{Z}\).

Answer:

\(x = 2k\pi,k\in\mathbb{Z}\)