Problem 78 - Algebra and Trigonometry

Question

Variable Stars Variable stars are ones whose brightness varies periodically. One of the most visible is R Leonis; its brightness is modeled by the function $$b(t)=7.9-2.1 \cos \left(\frac{\pi}{156} t\right)$$ where \(t\) is measured in days. (a) Find the period of R Leonis. (b) Find the maximum and minimum brightness. (c) Graph the function \(b\) .

Step-by-Step Solution

Answer

The period is 312 days; the max brightness is 10.0, and min is 5.8.

Step 1: Identify the function form

The brightness of R Leonis is modeled by the function given as \(b(t)=7.9-2.1 \cos \left(\frac{\pi}{156} t\right)\). This is a cosine function in the form \(A - C \cos(Bt)\), where \(A = 7.9\), \(C = 2.1\), and \(B = \frac{\pi}{156}\).

Step 2: Determine the period

The period of a cosine function \( \cos(kx) \) with coefficient \(k\) is given by \( \frac{2\pi}{k} \). Here, \(k = \frac{\pi}{156}\), so the period is \( \frac{2\pi}{\frac{\pi}{156}} = 2 \times 156 = 312 \). Thus, the period of R Leonis is 312 days.

Step 3: Calculate maximum brightness

The function \(b(t) = 7.9 - 2.1 \cos \left(\frac{\pi}{156} t\right)\) achieves its maximum when \(\cos\) is minimized. The minimum value of \(\cos(x)\) is \(-1\). Therefore, \(b_{\text{max}} = 7.9 - 2.1(-1) = 7.9 + 2.1 = 10.0\).

Step 4: Calculate minimum brightness

The function \(b(t)\) achieves its minimum when \(\cos\) is maximized. The maximum value of \(\cos(x)\) is \(1\). Therefore, \(b_{\text{min}} = 7.9 - 2.1(1) = 7.9 - 2.1 = 5.8\).

Step 5: Graph the function

Plot the function \(b(t) = 7.9 - 2.1\cos\left(\frac{\pi}{156} t\right)\) on a graph for a full period (0 to 312 days). The graph will oscillate between a maximum brightness of 10.0 and a minimum brightness of 5.8, repeating every 312 days.

Key Concepts

Cosine Function

The cosine function is one of the fundamental trigonometric functions. It represents the horizontal coordinate of the unit circle, fluctuating between -1 and 1 for any real number input. In the context of R Leonis' brightness model, the function is given by \(b(t) = 7.9 - 2.1 \cos \left(\frac{\pi}{156} t\right)\). This model utilizes the transformation of the basic cosine function to represent the observed brightness changes of the star.

The function form \(A - C \cos(Bt)\) indicates specific transformations:

• \(A = 7.9\): This represents the average or midline brightness.
• \(C = 2.1\): This indicates the amplitude, which is the maximum deviation from the midline.
• \(B = \frac{\pi}{156}\): This term affects the period of the function, changing how quickly the cosine graph repeats cycles.

By these modifications, the function maps the standard cosine wave to represent real-world data about R Leonis' brightness.

Function Period

The period of a trigonometric function is the interval over which the function completes one full cycle. For the cosine function \(\cos(kx)\), the period is calculated as \(\frac{2\pi}{k}\). In the case of R Leonis, where the modifying factor \(k\) is \(\frac{\pi}{156}\), the period becomes \(\frac{2\pi}{\left(\frac{\pi}{156}\right)} = 312\).

This means that the pattern of brightness increase and decrease of R Leonis repeats every 312 days. Understanding the period helps in predicting when the maximum and minimum brightness points occur throughout the observation timeframe.

Maximum and Minimum Values

The concept of maximum and minimum values in the function relates to the highest and lowest brightness levels R Leonis can achieve. In our model, these values are determined by the trigonometric characteristics of the cosine function:

• The maximum value of \(\cos(x)\) is 1. Therefore, the model's maximum brightness \(b_{\text{max}}\) occurs when \(\cos\left(\frac{\pi}{156} t\right) = -1\), giving \(b_{\text{max}} = 7.9 - 2.1(-1) = 10.0\).
• The minimum value of \(\cos(x)\) is -1. Therefore, the lowest brightness \(b_{\text{min}}\) occurs when \(\cos\left(\frac{\pi}{156} t\right) = 1\), resulting in \(b_{\text{min}} = 7.9 - 2.1(1) = 5.8\).

This oscillation between the calculated maxima and minima mimics the observable luminosity changes in variable stars like R Leonis.

Graphing Functions

Graphing functions, especially trigonometric ones like the cosine function, provides a visual representation of periodic behavior. For the brightness model of R Leonis, the function \(b(t) = 7.9 - 2.1 \cos \left(\frac{\pi}{156} t\right)\) should be graphed over the range of 0 to 312 days to depict one full cycle.

Key things to note when graphing include:

• Identify the midline. In this case, it's \(b = 7.9\), where the function centers.
• Mark the amplitude. The graph oscillates 2.1 units above and below the midline peak to maxima and minima of 10.0 and 5.8 respectively.
• Establish the period. The repeating pattern should be clear over the interval of 312 days, showcasing a recurrence of brightness levels.

This graphical representation helps to visualize and predict the behavior of the star's brightness over time, making it a crucial component in both analytical and observational astronomy.