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.