Problem 21 - Algebra 2

Question

Find the amplitude, if it exists, and period of each function. Then graph each function. \(y=\cot 5 \theta\)

Step-by-Step Solution

Answer

The amplitude does not exist for \( y = \cot(5\theta) \); the period is \( \frac{\pi}{5} \).

Step 1: Understanding the Cotangent Function

The general form of the cotangent function is \( y = a \cdot \cot(b\theta) \). In this function, \( y = \cot(5\theta) \), we have \( a = 1 \) and \( b = 5 \). The cotangent function, like all trigonometric functions except the sine and cosine, does not have an amplitude because it does not have a maximum or minimum value.

Step 2: Determining the Period of the Function

The period of \( \cot(b\theta) \) is given by \( \frac{\pi}{b} \). For \( y = \cot(5\theta) \), the value of \( b = 5 \). To find the period, we use the formula: \[ \text{Period} = \frac{\pi}{5} \] Therefore, the period of the function \( y = \cot(5\theta) \) is \( \frac{\pi}{5} \).

Step 3: Setting Up for Graphing

To graph \( y = \cot(5\theta) \), identify key points within one period \([0, \frac{\pi}{5}]\). The cotangent function has vertical asymptotes where the function is undefined at points that are integer multiples of \( \frac{\pi}{b} \). Since \( b = 5 \), the asymptotes will be at \( \theta = 0 \) and \( \theta = \frac{\pi}{5} \). The function is zero at the midpoint \( \theta = \frac{\pi}{10} \).

Step 4: Sketching the Graph

Plot the vertical asymptotes at \( \theta = 0 \) and \( \theta = \frac{\pi}{5} \). Mark the point where the function crosses the x-axis at \( \theta = \frac{\pi}{10} \), which is halfway between the asymptotes. The general shape of the cotangent function decreases from positive infinity to negative infinity as it moves from \( \theta = 0 \) to \( \theta = \frac{\pi}{5} \).

Key Concepts

Cotangent Function

The cotangent function, denoted as \( \cot \theta \), is one of the fundamental trigonometric functions. It is defined as the reciprocal of the tangent function, which means \( \cot \theta = \frac{1}{\tan \theta} \). This implies that the cotangent function is undefined wherever the tangent function is zero. These points of undefined value create vertical asymptotes on the graph because the function tends towards infinity.The equation for a cotangent function typically looks like \( y = a \cdot \cot(b\theta) \). In the specific function \( y = \cot(5\theta) \), the amplitude coefficient \( a \) is 1. However, unlike sine and cosine functions, the cotangent function does not have an amplitude. This is because it can stretch infinitely in the positive and negative directions without flattening out at the peaks and valleys as sine and cosine do.

Period of Trigonometric Functions

The concept of 'period' in trigonometric functions is crucial as it defines how frequently the function's pattern repeats over the horizontal axis. The period of the cotangent function is determined by the formula \( \frac{\pi}{b} \), where \( b \) is a coefficient affecting the frequency of the angle \( \theta \). For the function \( y = \cot(5\theta) \), the coefficient \( b = 5 \).Applying the period formula gives us:

• \( \text{Period} = \frac{\pi}{5} \)

This result indicates that the cotangent graph will complete one full cycle – transitioning from positive infinity to negative infinity and back again – over the interval \( [0, \frac{\pi}{5}] \). This shorter period, compared to the standard \( \pi \) of \( \cot \theta \), results from the angle being multiplied by 5, enhancing the frequency at which the function repeats its familiar waveform.

Graphing Trigonometric Functions

Graphing trigonometric functions such as the cotangent requires identifying key characteristics like vertical asymptotes, zeros, and the general direction of the curve. For \( y = \cot(5\theta) \), the graph starts by determining the locations of the vertical asymptotes, which occur where the function is undefined. These are at integer multiples of \( \frac{\pi}{b} \). Thus, for \( b = 5 \), the asymptotes lie at the boundaries of the period, specifically at \( \theta = 0 \) and \( \theta = \frac{\pi}{5} \). Right in the middle of this interval, where \( \theta = \frac{\pi}{10} \), the function has a zero crossing, as cotangent equals zero when \( \theta = \frac{\pi}{2} \).When sketching the graph:

• Start by drawing vertical dashed lines at the asymptotes \( \theta = 0 \) and \( \theta = \frac{\pi}{5} \).
• Mark the zero at \( \theta = \frac{\pi}{10} \).
• The curve decreases from positive infinity to negative infinity as it moves from \( \theta = 0 \) to \( \theta = \frac{\pi}{5} \).

Therefore, understanding these elements helps to accurately graph the function, revealing its unique wave-like shape.