on the interval 0 ≤ θ < 2π, give any solution of the equation tan θ = - 1

Explanation:

Step1: Recall tangent - angle relationship

We know that $\tan\theta=- 1$. The general solution of $\tan\theta = \tan\alpha$ is $\theta=n\pi+\alpha$, where $n\in\mathbb{Z}$. Since $\tan\left(-\frac{\pi}{4}\right)=-1$, the general solution is $\theta = n\pi-\frac{\pi}{4}$.

Step2: Find solutions in the given interval

For $n = 1$, $\theta=\pi-\frac{\pi}{4}=\frac{3\pi}{4}\approx2.36$. For $n = 2$, $\theta = 2\pi-\frac{\pi}{4}=\frac{7\pi}{4}\approx5.50$. In the interval $0\leq\theta<2\pi$, the solutions of $\tan\theta=-1$ are $\theta=\frac{3\pi}{4}$ and $\theta=\frac{7\pi}{4}$.

Answer:

$\theta=\frac{3\pi}{4},\frac{7\pi}{4}$