Problem 80 - 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

Period: 312 days; Max brightness: 5.8; Min brightness: 10.0.

Step 1: Understanding the Brightness Function

The given function for the brightness of R Leonis is \(b(t) = 7.9 - 2.1 \cos \left(\frac{\pi}{156} t\right)\). In this function, \(t\) represents time in days. The function is a cosine function that varies between two values, determined by the amplitude and midline.

Step 1: Determine the Period

The period of a cosine function \(a \cos(bt + c) + d\) is given by \(\frac{2\pi}{b}\). Here, \(b = \frac{\pi}{156}\). Thus, the period is \(\frac{2\pi}{\frac{\pi}{156}} = 312\). Therefore, the period of R Leonis is 312 days.

Step 2: Find Maximum Brightness

The maximum value of \(\cos(x)\) is 1. So, the maximum brightness, \(b_{max}\), occurs when \(\cos\left(\frac{\pi}{156} t \right) = 1\). Substitute this into the function: \(b_{max} = 7.9 - 2.1 \, \times\, 1 = 5.8\). Therefore, the maximum brightness is 5.8.

Step 3: Find Minimum Brightness

The minimum value of \(\cos(x)\) is -1. So, the minimum brightness, \(b_{min}\), occurs when \(\cos\left(\frac{\pi}{156} t \right) = -1\). Substitute this into the function: \(b_{min} = 7.9 - 2.1 \, \times\, (-1) = 10.0\). Hence, the minimum brightness is 10.0.

Step 4: Graphing the Brightness Function

To graph \(b(t) = 7.9 - 2.1 \cos \left(\frac{\pi}{156} t\right)\), plot the function over one period (0 to 312 days). The graph should oscillate between 5.8 and 10.0, with a period of 312 days. The midline of the function is at \(b = 7.9\), and it follows the shape of a cosine function.

Key Concepts

Period of a Function

The period of a function tells us how long it takes for the function to complete one full cycle before repeating. For trigonometric functions like cosine, the period is a crucial characteristic. It helps us understand the behavior of oscillating systems, such as the brightness of a variable star like R Leonis.

The formula to find the period of a cosine function, given in the form \(a \cos(bx + c) + d\), is \(\frac{2\pi}{b}\). In our function, \(b(t) = 7.9 - 2.1 \cos \left(\frac{\pi}{156} t\right)\), the value of \(b\) is \(\frac{\pi}{156}\).

• Compute the period by finding \(\frac{2\pi}{\frac{\pi}{156}}\).
• This simplifies to \(312\), indicating that R Leonis completes one cycle of brightness variation every 312 days.

Understanding the period can help astronomers and students predict when the star will appear brighter or dimmer from Earth, based on its regular cycle.

Maximum and Minimum Values

In trigonometric functions, maxima and minima represent the highest and lowest points the function can reach. For the brightness of R Leonis modeled by \(b(t) = 7.9 - 2.1 \cos \left(\frac{\pi}{156} t\right)\), these values indicate the brightest and dimmest appearances.

A cosine function ranges between -1 and 1. By placing these extremes into our function, we determine the maximum and minimum brightnesses.

• Maximum Brightness: Occurs when \(\cos(\theta) = 1\). Substituting, \(b_{max} = 7.9 - 2.1 \times 1 = 5.8\).
• Minimum Brightness: Occurs when \(\cos(\theta) = -1\). Substituting, \(b_{min} = 7.9 - 2.1 \times (-1) = 10.0\).

The star appears dimmest when \(b = 5.8\) and brightest when \(b = 10.0\). Recognizing these values helps in analyzing the star's luminosity changes over time.

Graphing Trigonometric Functions

Graphing trigonometric functions like the provided brightness function \(b(t) = 7.9 - 2.1 \cos \left(\frac{\pi}{156} t\right)\) provides a visual representation of the function's behavior. This is especially helpful in understanding how the brightness of R Leonis changes over time.

To graph, consider these key elements:

• Amplitudes and Midlines: The amplitude here is 2.1, which shows the height of the wave from the midline. The midline is at \(b = 7.9\).
• Maximum and Minimum Values: The graph oscillates between 5.8 (minimum) and 10.0 (maximum).
• Period: As calculated, one complete cycle spans 312 days.

First, plot the midline at 7.9. Map the maximum and minimum points at regular intervals, marking transitions from 5.8 to 10.0.

The visual graph will reveal the periodic nature of the brightness changes, aiding further understanding of oscillating patterns in real-world phenomena, like the luminosity variations of variable stars over time.