How to find asymptotes of a tangent function
Understand the Problem
The question is asking how to identify the asymptotes of the tangent function. Asymptotes are lines that the graph of a function approaches but never touches, and for the tangent function, this involves understanding its periodic nature and behavior at certain points.
Answer
The vertical asymptotes of the tangent function are located at $x = \frac{\pi}{2} + k\pi$, where $k \in \mathbb{Z}$.
Answer for screen readers
The vertical asymptotes of the tangent function are located at $x = \frac{\pi}{2} + k\pi$, where $k \in \mathbb{Z}$.
Steps to Solve
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Identify the tangent function The tangent function is defined as $y = \tan(x)$, which can be expressed as the ratio of the sine and cosine functions: $$ \tan(x) = \frac{\sin(x)}{\cos(x)} $$
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Determine where cosine is zero Asymptotes occur where the tangent function is undefined, which happens when the denominator (cosine) is zero. We need to solve for: $$ \cos(x) = 0 $$
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Find the values of x The cosine function equals zero at specific angles. For $x$ in radians, these points are: $$ x = \frac{\pi}{2} + k\pi, \quad k \in \mathbb{Z} $$ (where $k$ is any integer).
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List the vertical asymptotes From the previous step, we conclude that the vertical asymptotes for the tangent function are located at: $$ x = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, ... $$ and the negative counterparts: $$ x = -\frac{\pi}{2}, -\frac{3\pi}{2}, -\frac{5\pi}{2}, ... $$
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Understand the periodic nature The tangent function has a periodicity of $\pi$. Therefore, the vertical asymptotes repeat every $\pi$ units along the x-axis.
The vertical asymptotes of the tangent function are located at $x = \frac{\pi}{2} + k\pi$, where $k \in \mathbb{Z}$.
More Information
The tangent function has vertical asymptotes at intervals of $\pi$. This periodic nature influences the entire graph, causing it to replicate between each of these asymptotes. The tangent function is unique as its range is all real numbers, while it approaches infinity at the asymptotes.
Tips
- Students often forget to include the general solution for $k$, which leads to omitting an entire set of asymptotes.
- Some may incorrectly suggest that $\tan(x)$ has horizontal asymptotes, which it does not.
