Question

Asymptotes Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions. $$g(\theta)=\tan \left(\frac{\pi \theta}{10}\right)$$

Short answer

Question: Identify the vertical asymptotes of the function $$g(\theta) = \tan(\frac{\pi \theta}{10})$$. Answer: The vertical asymptotes of the function $$g(\theta)$$ are located at $$\theta = 5(2n + 1)$$, where $$n$$ is an integer.

Step-by-step solution

Find the function domain

To find the domain of the function, we need to find the values of $$\theta$$ for which the function is defined. Tangent is defined for all angles except for odd multiples of $$\frac{\pi}{2}$$. So we need to find when the argument of the tangent function, $$\frac{\pi \theta}{10}$$, is equal to an odd multiple of $$\frac{\pi}{2}$$, i.e., $$\frac{\pi \theta}{10} = \frac{(2n + 1)\pi}{2}$$ where $$n$$ is an integer.

Solve for theta

Now, we will solve for $$\theta$$ in the equation $$\frac{\pi \theta}{10} = \frac{(2n + 1)\pi}{2}$$ by isolating theta: $$\theta = \frac{(2n + 1)\cdot 10}{2}$$ Thus, $$\theta = 5(2n + 1)$$, where $$n$$ is an integer.

Identify vertical asymptotes

Since the tangent function becomes undefined at odd multiples of $$\frac{\pi}{2}$$, and we found that $$\theta = 5(2n + 1)$$ corresponds to these values, the vertical asymptotes of $$g(\theta)$$ are located at $$\theta = 5(2n + 1)$$, where $$n$$ is an integer.

Key concepts

Tangent Function

The tangent function, denoted as \( \tan(\theta) \), is an essential trigonometric function primarily used in mathematics to relate angles with side lengths in a right triangle. It arises by taking the ratio of the sine and cosine values for a given angle. Specifically, in a right triangle, the tangent function represents the ratio of the length of the opposite side to the length of the adjacent side relative to a particular angle.
- It's characterized by periodic behavior, with a period of \( \pi \), meaning that the pattern of the graph repeats every \( \pi \) units.
- This function has vertical asymptotes, or values where it is undefined, occurring at odd multiples of \( \frac{\pi}{2} \). During these points, the tangent function approaches positive or negative infinity.
Understanding the tendencies of the tangent function helps us predict its behavior and determine key features like asymptotes.

Domain of a Function

The domain of a function refers to the set of input values (usually represented as \( x \) or \( \theta \) in trigonometric contexts) for which the function is defined. For most mathematical functions, the domain includes all real numbers unless something specific restricts this.
For the tangent function described as \( g(\theta) = \tan \left(\frac{\pi \theta}{10}\right) \), the domain is constrained because the tangent function is undefined at points where its denominator (cosine) equals zero, which happens at odd multiples of \( \frac{\pi}{2} \).
In solving \( \frac{\pi \theta}{10} = \frac{(2n + 1)\pi}{2} \), we find the specific points \( \theta = 5(2n + 1) \) that are excluded from the domain, where \( n \) is an integer. This step is crucial to identify where the function fails to produce a valid number and leads to finding vertical asymptotes.

Graphing Utility

A graphing utility is a helpful technological tool—like graphing calculators, software, or apps—that visualizes functions. It helps learners see the behavior of a function, especially in trigonometric and calculus contexts, providing graphics that illustrate concepts like intercepts, periodicity, and asymptotes.
When examining a function like \( g(\theta) = \tan \left(\frac{\pi \theta}{10}\right) \), inputting it into a graphing utility allows one to see the vertical asymptotes as the places where the graph shoots up or down infinitely.

• It visually simplifies complex algebra into identifiable patterns, making it easier to predict further behavior of the function.
• Using a graphing utility can confirm analytic work by displaying these patterns and checking for possible errors in calculation.

This visual approach can help deepen understanding and support learning by presenting abstract mathematical concepts in a more tangible form.

Trigonometric Functions

Trigonometric functions are foundational in trigonometry, a branch of mathematics dealing with angles and their relationships with lengths of triangles. The major functions include sine, cosine, and tangent, among others, each uniquely behaving to describe characteristics of angles and arcs.
- They are periodic, meaning they repeat their values in regular intervals. - These functions have applications from pure mathematics to practical fields such as physics, engineering, and architecture due to their ability to model oscillatory phenomena.
Specifically, the tangent function emerges as critical due to its unique feature of possessing vertical asymptotes, which no other basic trigonometric function has in the same manner. Understanding trigonometric functions like '\( \tan \)' is crucial for solving problems related to rotations, waves, and circular motion.
With repetitive patterns and significant flexibility, they serve as versatile tools for translating angles into practical information, bridging the gap between geometric intuition and algebraic computation.