Tangent and Cotangent Graphs

From the definition of the tangent and cotangent functions, we have

$$\tan( \theta)= \frac{\sin(\theta)}{\cos(\theta)},\quad \cot( \theta)= \frac{\cos(\theta)}{\sin(\theta)}.$$

Thus, $\tan(\theta)$ is not defined for values of $\theta$ such that $\cos(\theta) = 0$. Now, consider the graph of $\cos (\theta)$:

From this graph, we see that $\cos(\theta) = 0$ when $\theta = \frac{\pi}{2} + k\pi$ for any integer $k$. This implies that the tangent function has vertical asymptotes at these values of $\theta$.

Does the tangent function approach positive or negative infinity at these asymptotes? As $\theta$ approaches $\frac{\pi}{2}$ from below $\big(\theta$ takes values less than $\frac{\pi}{2}$ while getting closer and closer to $\frac{\pi}{2}\big),$ $\sin (\theta)$ takes positive values that are closer and closer to $1$, while $\cos (\theta)$ takes positive values that are closer and closer to $0$. This shows $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$ is positive and approaches infinity, so $\tan(\theta)$ has a positive vertical asymptote as $\theta \rightarrow \frac{\pi}{2}$ from below. By a similar analysis, as $\theta$ approaches $\frac{\pi}{2}$ from above $\big(\theta$ takes values larger than $\frac{\pi}{2}$ while getting closer and closer to $\frac{\pi}{2}\big),$ $\sin (\theta)$ takes positive values that are closer and closer to $1$, while $\cos (\theta)$ takes negative values that are closer and closer to $0$. This shows $\tan(\theta)$ has a negative vertical asymptote as $\theta \rightarrow \frac{\pi}{2}$ from above. The following shows the graph of tangent for the domain $0 \leq \theta \leq 2\pi$:

The graph of tangent over its entire domain is as follows:

Similarly, $\cot(\theta)$ is not defined for values of $\theta$ such that $\sin(\theta) = 0$. From the graph of $\sin (\theta),$ we see that $\sin(\theta) = 0$ when $\theta = 0 + k\pi$ for any integer $k$, which implies that the cotangent function has vertical asymptotes at these values of $\theta:$

Observe that from the definition of tangent and cotangent, we obtain the following relationship between the tangent and cotangent functions:

$$\tan(\theta) = \frac{\sin (\theta)}{\cos (\theta)} = \frac{1} {\ \ \frac{\cos (\theta)}{\sin (\theta)}\ \ } = \frac{1}{\cot(\theta)}.$$

Indeed, we can see that in the graphs of tangent and cotangent, the tangent function has vertical asymptotes where the cotangent function has value 0 and the cotangent function has vertical asymptotes where the tangent function has value 0.

The tangent and cotangent graphs satisfy the following properties:

• range: $(-\infty, \infty)$
• period: $\pi$
• both are odd functions.

From the graphs of the tangent and cotangent functions, we see that the period of tangent and cotangent are both $\pi$. In trigonometric identities, we will see how to prove the periodicity of these functions using trigonometric identities.

What values of $\theta$ in the interval $[0, \pi]$ satisfy $\tan(\theta) = \cot(\theta)?$ Can we see this from the graphs of the tangent and cotangent functions?

---

We would like to find values of $\theta$ such that $\tan(\theta) = \cot(\theta) = \frac{1}{\tan(\theta)}$, i.e. $\big( \tan (\theta) \big)^2 = 1$. This is satisfied for $\tan \theta = \pm 1$, or $\theta = \frac{\pi}{4}, \frac{3\pi}{4}$.

We can also see from the graphs of tangent and cotangent that the points of intersection of the two graphs in the domain $[0,\pi]$ are $\big( \frac{\pi}{4}, 1 \big)$ and $\big( \frac{3\pi}{4}, -1 \big).\ _\square$