Problem 45 - Algebra and Trigonometry

Question

People who believe in biorhythms claim that there are three cycles that rule our behavior - the physical, emotional, and mental. Each is a sine function of a certain period. The function for our emotional fluctuations is $$ E=\sin \frac{\pi}{14} t $$ where \(t\) is measured in days starting at birth. Emotional fluctuations, \(E,\) are measured from \(-1\) to \(1,\) inclusive, with 1 representing peak emotional well-being, \(-1\) representing the low for emotional well-being, and 0 representing feeling neither emotionally high nor low. a. Find \(E\) corresponding to \(t=7,14,21,28,\) and \(35 .\) Describe what you observe. b. What is the period of the emotional cycle?

Step-by-Step Solution

Answer

The values of \(E\) corresponding to \(t = 7, 14, 21, 28,\) and \(35\) are 1, 0, -1, 0, and 1 respectively, showing a pattern of emotional fluctuation. The period of the emotional cycle is 28 days.

Step 1: Evaluate \(E\) for Given Values

We're asked to find the values of \(E\) corresponding to \(t = 7, 14, 21, 28,\) and \(35\). We'll substitute these values into \(E = \sin\frac{{\pi}}{{14}}t\). For \(t = 7\), \(E = \sin\left(\frac{{\pi}}{{14}}\times 7\right) = \sin\left(\frac{{\pi}}{{2}}\right) = 1\). Similarly, calculate the values for remaining \(t\)s.

Step 2: Describe Observations

On calculating \(E\) for the given values of \(t\), we'll find that \(E\) cycles from 1 to -1 and back to 1. This corresponds to the individual's emotional cycle, oscillating from peak emotional well-being (1), to low emotional well-being (-1), and back to neutral (0).

Step 3: Determine the Period of the Emotional Cycle

The period of a sine function is the value at which the function completes one full cycle. Given the function \(E = \sin\frac{{\pi}}{{14}}t\), the period can be found as \(2\pi / \frac{{\pi}}{{14}} = 28\) days. This means the emotional cycle repeats every 28 days.

Key Concepts

Sine Function

The sine function is a fundamental concept in trigonometry and is often used to model periodic phenomena, like waves or cycles. The general form of the sine function is \( y = \sin(kt) \), where \( k \) determines the frequency of the cycle. The sine wave oscillates between -1 and 1.

In the context of biorhythms, the emotional cycle is represented as a sine function \( E = \sin(\frac{\pi}{14} t) \). Here, \( t \) measures the number of days since birth. By plugging in different values of \( t \), we can determine the emotional state on any given day:

• \( E = 1 \) represents peak emotional well-being.
• \( E = -1 \) indicates the emotional low point.
• \( E = 0 \) signifies a neutral emotional state.

The simplicity of the sine function makes it ideal for modeling regular and predictable fluctuations, such as those proposed in the biorhythm theory.

Emotional Fluctuations

Emotional fluctuations describe the changes in emotional states over time. In biorhythm theory, these are thought to follow predictable patterns, modeled by mathematical functions like the sine function. This approach attempts to quantify how emotional well-being peaks and troughs over a specific time frame, dictated by the sine wave's oscillation.

When considering the function \( E = \sin(\frac{\pi}{14} t) \):

• The peaks (\

Emotional Fluctuations

Emotional fluctuations describe the changes in emotional states over time. In biorhythm theory, these are thought to follow predictable patterns, modeled by mathematical functions like the sine function. This approach attempts to quantify how emotional well-being peaks and troughs over a specific time frame, dictated by the sine wave's oscillation.When considering the function \( E = \sin(\frac{\pi}{14} t) \):- The peaks (

Emotional Fluctuations

Emotional fluctuations describe the changes in emotional states over time. In biorhythm theory, these are thought to follow predictable patterns, modeled by mathematical functions like the sine function. This approach attempts to quantify how emotional well-being peaks and troughs over a specific time frame, dictated by the sine wave's oscillation.When considering the function \( E = \sin(\frac{\pi}{14} t) \):- The peaks (

Emotional Fluctuations

Emotional fluctuations describe the changes in emotional states over time. In biorhythm theory, these are thought to follow predictable patterns, modeled by mathematical functions like the sine function. This approach attempts to quantify how emotional well-being peaks and troughs over a specific time frame, dictated by the sine wave's oscillation.When considering the function \( E = \sin(\frac{\pi}{14} t) \):- The peaks (