Problem 32 - Algebra 2

Question

Writing Explain how you can find the equations of the asymptotes of \(y=\tan b \theta\)

Step-by-Step Solution

Answer

The equations of the asymptotes of the function \(y=\tan b \theta\) are given by \(\theta = \frac{(2n+1)\pi}{2b}\), where \(n\) is an integer.

Step 1: Understanding the tangent function

The tangent function can be denoted as \(y = \tan b\theta = \frac{\sin b\theta}{\cos b\theta}\). The function is not defined where \(\cos b\theta = 0\). This undefined behaviour happens periodically, creating asymptotes in the graph of tangent.

Step 2: Identifying when the cosine function is 0.

The cosine function becomes 0 at odd multiples of \(\pi/2\), thus, we consider \(\cos b\theta = 0\). Solving this equation leads us to \(b\theta = \frac{(2n+1)\pi}{2}\), where \(n\) is an integer.

Step 3: Find the general equation of the asymptotes

From Step 2, we can find the value of \(\theta\) by doing \(b\theta = \frac{(2n+1)\pi}{2}\) to obtain \(\theta = \frac{(2n+1)\pi}{2b}\). These are the equations of the asymptotes since the function has vertical asymptotes at these \(\theta\)s

Key Concepts

Tangent Function

The tangent function is a fundamental trigonometric function, indicated as \( y = \tan b\theta \). It is the ratio of the sine and cosine functions:

• \( y = \frac{\sin b\theta}{\cos b\theta} \)

This function reveals many interesting characteristics due to its dependence on sine and cosine. It behaves as the sine wave divided by the cosine wave, which has significant implications.
Firstly, the tangent function will become undefined whenever the cosine of the angle is zero. When this occurs, the denominator becomes zero, leading to undefined values.
This undefined behavior creates vertical asymptotes in the graph at these points, which we'll further explore. The properties of sine and cosine give the tangent its unique shape.

Periodicity

Periodicity is a core property of trigonometric functions, whereby they repeat their values at regular intervals. For the tangent function, its periodic nature means it has a repeating pattern

• The tangent function repeats every \( \pi \) radians.
• This repeat interval is known as the period.

Writing it mathematically, if \( \theta \) is an angle, then \( \tan (\theta + n\pi) = \tan \theta \) for any integer \( n \). This characteristic is vital for understanding the function's graph.
Due to this repetition every \( \pi \) radians, determining where asymptotes occur helps illustrate the periodicity. It directly ties into where the cosine function, part of the tangent expression, reaches zero repeatedly.

Undefined Behavior

Undefined behavior occurs in functions when certain operations aren't determinable, often leading to infinities or unallowed values.

• In the tangent function, undefined points arise when \( \cos b\theta = 0 \).
• This occurs because division by zero is not possible.

The cosine of an angle becomes zero at odd multiples of \( \frac{\pi}{2} \). Hence, these values disrupt the tangent's definedness, resulting in vertical asymptotes.
Mathematically, this is expressed as \( b\theta = \frac{(2n+1)\pi}{2} \).
Understanding where these undefined points occur lets us determine the pattern of the tangent's vertical asymptotes, crucial for sketching its graph.

Tangent Graph

The tangent graph is unique among trigonometric graphs due to its pattern of asymptotes and periodic shape.

• It characteristically spans from negative infinity to positive infinity between consecutive asymptotes.
• The graph appears as flowing upwards from left to right in each period.

When plotting, the locations of the vertical asymptotes are pivotal, marking points where the graph does not exist. These are regularly spaced along the \( \theta \)-axis based on the equation \( \theta = \frac{(2n+1)\pi}{2b} \).
As these asymptotes, derived from the cosine zero points, dictate the graph's discontinuities, evaluating them assists learners in accurately sketching the graph.
Therefore, comprehending the influence of undefined behavior on the tangent graph helps visualize its wave-like, repeating nature.