Solve each of the following equations for theta, where θ ϵ [-2π, 2π]: tan θ = -1 

To solve the equation \( \tan(\theta) = -1 \) for \( \theta \) within the interval \( \theta \in [-2\pi, 2\pi] \), we need to follow these steps:

### Step 1: Understand the General Solution for \( \tan(\theta) = -1 \)
The tangent function \( \tan(\theta) \) equals \( -1 \) at specific angles in the unit circle. Tangent is negative in the second and fourth quadrants, where:

- \( \theta = \frac{3\pi}{4} \) in the second quadrant (reference angle \( \frac{\pi}{4} \)).
- \( \theta = -\frac{\pi}{4} \) in the fourth quadrant (reference angle \( \frac{\pi}{4} \)).

However, since the tangent function has a period of \( \pi \), the general solution for \( \tan(\theta) = -1 \) is given by:
\[
\theta = \frac{3\pi}{4} + n\pi \quad \text{where} \, n \, \text{is an integer.}
\]

### Step 2: Find Specific Solutions in the Interval \( [-2\pi, 2\pi] \)
Now, we need to find the values of \( \theta \) within the interval \( [-2\pi, 2\pi] \). We start from the general solution and add multiples of \( \pi \) until we exhaust all possibilities in the interval.

- Starting with \( \theta = \frac{3\pi}{4} \):
  - \( \frac{3\pi}{4} \)
  - \( \frac{3\pi}{4} - \pi = -\frac{\pi}{4} \)
  - \( \frac{3\pi}{4} + \pi = \frac{7\pi}{4} \)
  - \( \frac{3\pi}{4} - 2\pi = -\frac{5\pi}{4} \)
  - \( \frac{3\pi}{4} + 2\pi = \frac{11\pi}{4} \) (This is outside the interval, so it's excluded.)

### Step 3: List the Solutions
The solutions to the equation \( \tan(\theta) = -1 \) in the interval \( [-2\pi, 2\pi] \) are:
\[
\theta = -\frac{5\pi}{4}, -\frac{\pi}{4}, \frac{3\pi}{4}, \frac{7\pi}{4}
\]

### Final Answer:
The values of \( \theta \) that satisfy \( \tan(\theta) = -1 \) in the interval \( [-2\pi, 2\pi] \) are:
\[
\theta = -\frac{5\pi}{4}, -\frac{\pi}{4}, \frac{3\pi}{4}, \frac{7\pi}{4}
\]

Would you like further clarification or have any additional questions? Here are some related questions:

1. Solve \( \tan(\theta) = 1 \) in the interval \( [-2\pi, 2\pi] \).
2. Find all solutions for \( \sin(\theta) = -\frac{1}{2} \) in \( [-2\pi, 2\pi] \).
3. Solve \( \cos(\theta) = 0 \) in \( [-2\pi, 2\pi] \).
4. Determine the general solution for \( \tan(\theta) = 0 \).
5. Solve \( \tan(2\theta) = 1 \) for \( \theta \in [-2\pi, 2\pi] \).

**Tip:** For trigonometric equations, consider the reference angle and the quadrants where the function takes the desired sign.

Learn how to solve the trigonometric equation tan(theta) = -1 within the interval [-2pi, 2pi]. This problem explores the behavior of the tangent function, its values in specific quadrants of the unit circle, and how to find solutions within a given interval. Suitable for high school students.

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