*x*=3*π*​+2*kπ*,*x*=−3*π*​+2*kπ*,*x*=*π*+2*kπ*,*k*∈Z SIMPLIFY

Let's simplify the solutions:

Given the solutions:
\[
x = 3\pi + 2k\pi, \quad x = -3\pi + 2k\pi, \quad x = \pi + 2k\pi, \quad k \in \mathbb{Z}
\]

### Simplification

1. **First solution:** \( x = 3\pi + 2k\pi \)
   \[
   x = \pi(3 + 2k)
   \]
   This can be rewritten as \( x = \pi(1 + 2(k+1)) \). Since \( k+1 \) is just another integer, we can denote it as \( m \) (where \( m = k+1 \)). Therefore:
   \[
   x = \pi + 2m\pi, \quad m \in \mathbb{Z}
   \]
   This is equivalent to \( x = \pi + 2k\pi \), where \( k \in \mathbb{Z} \).

2. **Second solution:** \( x = -3\pi + 2k\pi \)
   \[
   x = \pi(-3 + 2k)
   \]
   This can be rewritten as \( x = -\pi + 2(k-1)\pi \). Again, since \( k-1 \) is just another integer, denote it as \( n \) (where \( n = k-1 \)). Therefore:
   \[
   x = -\pi + 2n\pi, \quad n \in \mathbb{Z}
   \]
   This is equivalent to \( x = -\pi + 2k\pi \), where \( k \in \mathbb{Z} \).

3. **Third solution:** \( x = \pi + 2k\pi \) is already in its simplest form.

### Final Simplified Solutions
The three distinct simplified solutions are:
\[
x = \pi + 2k\pi, \quad x = -\pi + 2k\pi, \quad x = \pi + 2k\pi, \quad k \in \mathbb{Z}
\]

Notice that the first and third solutions are identical, so they can be combined, giving us the final unique solutions:
\[
x = \pi + 2k\pi \quad \text{and} \quad x = -\pi + 2k\pi, \quad k \in \mathbb{Z}
\]

Would you like to explore further details or have any specific questions? Here are five related questions:

1. How can you determine when different trigonometric equation solutions are equivalent?
2. What other methods exist to simplify solutions for trigonometric equations?
3. How does the general solution of trigonometric equations relate to the unit circle?
4. Can trigonometric solutions be represented graphically for better understanding?
5. How can periodicity of trigonometric functions affect the solution set?

**Tip:** Always look for overlapping solutions when simplifying trigonometric equations, as they may reduce the number of distinct solutions.

Learn how to simplify trigonometric equations with multiple solutions, including examples like x = 3π + 2kπ, x = -3π + 2kπ, and x = π + 2kπ. Understand the periodic nature of trigonometric functions and how to identify equivalent solutions. Suitable for high school students studying trigonometry.

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