Problem 11 - Algebra 2

Question

Identify the period and tell where two asymptotes occur for each function. $$ y=\tan 5 \theta $$

Step-by-Step Solution

Answer

The period of the function \(y=\tan 5\theta\) is \(\pi/5\). Asymptotes occur where \(\theta = \pi/10 + k*\pi/5\), for all values of k as an integer.

Step 1: Identify Period

First, remember the period of the standard tangent function, \(y=\tan\theta\), is \(\pi\). If there's a coefficient, say n, as in \(y=\tan n\theta\), the period becomes \(\pi/n\). In this case, n=5, therefore, the period of \(y=\tan 5\theta\) is \(\pi/5\).

Step 2: Identify Asymptotes

Asymptotes for the standard tangent function, occur at \(-\pi/2\) and \(\pi/2\), and then repeat every \(\pi\) units. Because the period here is changed to \(\pi/5\), asymptotes occur at \(-\pi/(2*5)\) and \(\pi/(2*5)\), and then repeat every \(\pi/5\) units. Therefore, asymptotes occur where \(\theta = \pi/(10) + k*(\pi/5)\), where k is an integer.

Key Concepts

Period of a Function

Understanding the period of a function is crucial when dealing with periodic or cyclical functions. The period of a function is the interval over which the function's values repeat. For trigonometric functions, this concept is particularly important. Consider the standard tangent function, \( y = \tan\theta \), which has a period of \( \pi \). This means that the pattern of the tangent function repeats every \( \pi \) units.

But things change when we introduce a coefficient to the angle in the tangent function. When you see an equation like \( y = \tan n\theta \), the period of the function will be affected. The formula to find the new period becomes \( \pi / n \). Let's take the function \( y = \tan 5\theta \). Here, \( n \) equals 5, which means that the period becomes \( \pi/5 \). In essence, the cycle completes itself in a reduced interval of \( \pi/5 \).

It's important to keep in mind that altering the period essentially compresses or stretches the function on the horizontal axis. This is a fundamental transformation when dealing with trigonometric functions and understanding this will help you better analyze such functions.

Tangent Function

The tangent function is one of the six fundamental trigonometric functions and is defined as the ratio of the sine and cosine functions: \( \tan\theta = \frac{\sin\theta}{\cos\theta} \). This function can be visualized on the unit circle and has some distinct characteristics that set it apart from other trigonometric functions.

- **Characteristics of the Tangent Function:** The tangent function is periodic, oscillating with regular intervals. Unlike sine and cosine, the tangent function does not have an amplitude (maximum height), as it can take any real number value. - **Behavior of the Graph:** A key aspect of the tangent function's graph is that it includes asymptotes, where the function is undefined. This occurs when the cosine angle is zero, leading to division by zero. - **Symmetry:** The tangent function exhibits an odd symmetry, meaning it is symmetric about the origin, specifically \( \tan(-\theta) = -\tan(\theta) \).

Understanding these properties helps in sketching the function accurately and predicting its behavior as \( \theta \) changes. Such insights are crucial when solving trigonometric problems involving transformations.

Asymptotes

Asymptotes are lines that a graph approaches but never actually reaches. They are important in understanding the behavior of functions, particularly the tangent function. For the standard tangent function, \( y = \tan\theta \), vertical asymptotes occur at values of \( \theta \) where the function becomes undefined, specifically when \( \cos\theta = 0 \). This happens at \( \theta = -\pi/2, \pi/2, \) and every integer multiple of \( \pi \).

For transformations, like \( y = \tan 5\theta \), the asymptotes change according to the altered periodic interval. As the period is \( \pi/5 \), the vertical asymptotes occur more frequently. They can be determined using the equation for the asymptotes: \( \theta = \frac{\pi}{10} + k(\frac{\pi}{5}) \), where \( k \) is an integer. These lines never intersect the graph but approach at infinitesimally close margins, suggesting the limits of the tangent function.

Grasping the concept of asymptotes helps in sketching functions with precision, and understanding asymptotic behavior is invaluable when diving deeper into calculus and limits.