To solve the equation [tex]\(\tan(2x) - \tan(x) = 0\)[/tex] in the interval [tex]\([0, 2\pi)\)[/tex], follow these steps:
1. Understand the Equation:
The equation is given as:
[tex]\[
\tan(2x) - \tan(x) = 0
\][/tex]
This can be rewritten as:
[tex]\[
\tan(2x) = \tan(x)
\][/tex]
2. Use a Trigonometric Identity:
Recall that the tangent function is periodic with a period of [tex]\(\pi\)[/tex]. This means for any angle [tex]\(\theta\)[/tex], we have:
[tex]\[
\tan(\theta) = \tan(\theta + n\pi) \quad \text{for} \quad n \in \mathbb{Z}
\][/tex]
Applying this to our equation, we get:
[tex]\[
2x = x + n\pi
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
Rearrange to solve for [tex]\(x\)[/tex]:
[tex]\[
2x - x = n\pi
\][/tex]
[tex]\[
x = n\pi
\][/tex]
4. Determine Possible Solutions Within the Interval [tex]\( [0, 2\pi) \)[/tex]:
We need to find values of [tex]\(x\)[/tex] that lie within the interval [tex]\([0, 2\pi)\)[/tex]. The general solution [tex]\(x = n\pi\)[/tex] where [tex]\(n \in \mathbb{Z}\)[/tex] dictates which specific values need to be considered:
Start with [tex]\(n = 0\)[/tex]:
[tex]\[
x = 0\pi = 0
\][/tex]
For [tex]\( n = 1 \)[/tex]:
[tex]\[
x = 1\pi = \pi
\][/tex]
For [tex]\( n = 2 \)[/tex]:
[tex]\[
x = 2\pi
\][/tex]
But [tex]\(2\pi\)[/tex] is not within the interval [tex]\([0, 2\pi)\)[/tex] and lies at the boundary.
5. List the Valid Solutions:
Thus, the solutions to the equation [tex]\(\tan(2x) = \tan(x)\)[/tex] in the interval [tex]\([0, 2\pi)\)[/tex] are:
[tex]\[
x = 0, \pi
\][/tex]
Rechecking these within the context confirms their validity for the given interval.
6. Filtered Solutions:
For an inclusive strictly less than [tex]\(2\pi\)[/tex], we accept 0. Hence the valid solution practically is:
[tex]\[
[0]
\][/tex]
So, the solution set for [tex]\(\tan(2x) - \tan(x) = 0\)[/tex] within the interval [tex]\([0, 2\pi)\)[/tex] is [tex]\([0]\)[/tex].
