2. does the graph of tan(θ) have other vertical asymptotes? explain how you know.

Explanation:

Step1: Recall tangent - cosine relationship

The tangent function is defined as $\tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)}$. Vertical asymptotes occur where the function is undefined. A rational function is undefined when the denominator is zero.

Step2: Find when cosine is zero

We know that $\cos(\theta) = 0$ when $\theta=\frac{\pi}{2}+n\pi$, where $n\in\mathbb{Z}$ (the set of all integers).

Step3: Determine vertical asymptotes

Since $\tan(\theta)$ is undefined when $\cos(\theta)=0$, the graph of $y = \tan(\theta)$ has vertical asymptotes at $\theta=\frac{\pi}{2}+n\pi$, $n\in\mathbb{Z}$. There are infinitely many such values of $\theta$ (for different integer values of $n$), so there are other vertical asymptotes.

Answer:

Yes, the graph of $y = \tan(\theta)$ has other vertical asymptotes. The vertical asymptotes occur at $\theta=\frac{\pi}{2}+n\pi$, where $n\in\mathbb{Z}$. This is because $\tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)}$ and the function is undefined when $\cos(\theta) = 0$, and $\cos(\theta)=0$ at $\theta=\frac{\pi}{2}+n\pi$ for all integers $n$.