Solve the trigonometric equation for all values 0, is less than, x, is less than or equal to, 2, pi0<*x*≤2*π*.

tangent, 2, x, plus, 1, equals, 0

tan2*x*+1=0

The trigonometric equation to solve is:

\[
\tan(2x) + 1 = 0
\]

We need to find all solutions for \( x \) within the interval \( 0 < x \leq 2\pi \).

### Step 1: Isolate the tangent function
Start by isolating the tangent function:

\[
\tan(2x) = -1
\]

### Step 2: Find the general solution for \( 2x \)
The tangent function equals \(-1\) at the following angles:

\[
2x = \frac{3\pi}{4} + n\pi \quad \text{where } n \text{ is an integer}
\]

### Step 3: Solve for \( x \)
To find \( x \), divide both sides by 2:

\[
x = \frac{3\pi}{8} + \frac{n\pi}{2}
\]

### Step 4: Determine \( n \) such that \( 0 < x \leq 2\pi \)
Substitute into the interval \( 0 < x \leq 2\pi \):

\[
0 < \frac{3\pi}{8} + \frac{n\pi}{2} \leq 2\pi
\]

Now, solve for \( n \) by considering both ends of the inequality.

1. **Lower bound:**
   \[
   \frac{3\pi}{8} + \frac{n\pi}{2} > 0
   \]
   Simplifying, this is always true for any \( n \).

2. **Upper bound:**
   \[
   \frac{3\pi}{8} + \frac{n\pi}{2} \leq 2\pi
   \]
   Subtract \( \frac{3\pi}{8} \) from both sides:
   \[
   \frac{n\pi}{2} \leq 2\pi - \frac{3\pi}{8} = \frac{13\pi}{8}
   \]
   Multiply by 2:
   \[
   n\pi \leq \frac{13\pi}{4}
   \]
   Divide by \( \pi \):
   \[
   n \leq \frac{13}{4} = 3.25
   \]
   So, \( n \) can be 0, 1, 2, or 3.

### Step 5: Find the specific solutions
Substitute these values of \( n \) into \( x = \frac{3\pi}{8} + \frac{n\pi}{2} \):

- For \( n = 0 \): 
  \[
  x = \frac{3\pi}{8}
  \]

- For \( n = 1 \): 
  \[
  x = \frac{3\pi}{8} + \frac{\pi}{2} = \frac{7\pi}{8}
  \]

- For \( n = 2 \): 
  \[
  x = \frac{3\pi}{8} + \frac{\pi}{2} \times 2 = \frac{11\pi}{8}
  \]

- For \( n = 3 \): 
  \[
  x = \frac{3\pi}{8} + \frac{\pi}{2} \times 3 = \frac{15\pi}{8}
  \]

### Final Answer:
The solutions to the equation \( \tan(2x) + 1 = 0 \) in the interval \( 0 < x \leq 2\pi \) are:

\[
x = \frac{3\pi}{8}, \quad \frac{7\pi}{8}, \quad \frac{11\pi}{8}, \quad \frac{15\pi}{8}
\]

Would you like more details, or do you have any other questions?

### Related Questions:
1. What are the solutions to \( \tan(kx) + c = 0 \) for different constants \( k \) and \( c \)?
2. How do you solve the equation \( \sin(2x) = -1 \) in the interval \( 0 < x \leq 2\pi \)?
3. What is the period of the function \( \tan(2x) \) and how does it affect the number of solutions?
4. How can you verify trigonometric solutions using a unit circle?
5. What changes if the equation involves \( \cot(2x) \) instead of \( \tan(2x) \)?

### Tip:
When working with trigonometric equations, always consider the periodicity of the function to determine all possible solutions within the given interval.

Learn how to solve the trigonometric equation tan(2x) + 1 = 0 within the interval 0 < x ≤ 2π. Explore the step-by-step solution process, including isolating the tangent function, finding general solutions, and determining specific values. Suitable for advanced high school students and beyond.

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