Problem 10 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

Why are all the points of the graph of \(y=\log _{10}(\sin (x))\) on or below the X-axis? Why are there no points of the graph with \(x\) -coordinates between \(\pi\) and \(2 \pi ?\)

Step-by-Step Solution

Answer

The graph is on or below the X-axis because \(\log_{10}(\sin(x))\leq 0\) for positive values of \(\sin(x)\); there are no points for \(x\) between \(\pi\) and \(2\pi\) since \(\sin(x)\) is negative or zero.

Step 1: Understanding Logarithm

The function given is \(y = \log_{10}(\sin(x))\). Logarithm of a number is defined only when the number is positive. Thus, \(\sin(x)\) must be greater than zero for \(y\) to be defined.

Step 2: Interval for Sine Function Positivity

The sine function \(\sin(x)\) is positive only in the interval from \(0\) to \(\pi\), as sine values become negative from \(\pi\) to \(2\pi\). This means \(x\) must lie within \(0 < x < \pi\) for \(\log_{10}(\sin(x))\) to be defined and real.

Step 3: Logarithm Values Relative to X-axis

Since the logarithm of numbers between 0 and 1 is negative, \(\log_{10}(\sin(x))\) is less than or equal to zero for \(0 < x < \pi\). Hence, all points of the graph will be on or below the X-axis.

Step 4: Absence of Graph Points Between \(\pi\) and \(2\pi\)

For \(x\) in the interval \(\pi < x < 2\pi\), \(\sin(x)\) is negative, and as a result, \(\log_{10}(\sin(x))\) is not defined because logarithms for negative numbers do not exist. Thus, there are no points of the graph with \(x\)-coordinates between \(\pi\) and \(2\pi\).

Key Concepts

Sine Function

The sine function, represented as \( \sin(x) \), is one of the fundamental trigonometric functions. It reflects the vertical position of a point on the unit circle as it moves along its circumference. This function cycles through values ranging from -1 to 1 as the angle \( x \) varies over an interval. Here are some quick features of the sine function:

• The sine of 0 degrees (or 0 radians) is 0, and it reaches its maximum value, 1, at 90 degrees (or \( \pi/2 \) radians).
• As \( x \) continues to increase, \( \sin(x) \) decreases, becoming 0 again at 180 degrees (or \( \pi \) radians).
• Beyond \( \pi \), the sine function takes negative values, hitting -1 at 270 degrees (or \( 3\pi/2 \) radians) and returning to 0 at 360 degrees (or \( 2\pi \) radians).

These characteristics are essential when analyzing functions that involve the sine function, such as logarithmic functions, which require positive inputs.

Graph Analysis

To analyze the graph of a mathematical function, we observe how changes in the input (\( x \)) affect the output (\( y \)). For the function \( y = \log_{10}(\sin(x)) \), analysis involves understanding both the behavior of the sine function and the properties of logarithms.Logarithmic functions, like \( y = \log_{10}(x) \), are defined only for positive \( x \). Hence, for \( \log_{10}(\sin(x)) \) to be valid, \( \sin(x) \) must be positive. This occurs when \( x \) is in the interval from 0 to \( \pi \). Within this interval, the value of \( \sin(x) \) lies between 0 and 1, making \( \log_{10}(\sin(x)) \) negative. As a result, the graph will be below the X-axis.Additionally, since \( \sin(x) \) becomes negative between \( \pi \) and \( 2\pi \), \( \log_{10}(\sin(x)) \) becomes undefined for this range. Therefore, no graph points exist for \( x \) values between \( \pi \) and \( 2\pi \). Understanding these behaviors is key to accurately interpreting the graph.

Trigonometric Functions

Trigonometric functions are a crucial component in understanding angles and periodic phenomena in mathematics. There are six primary trigonometric functions, and the sine function is one of them, alongside cosine, tangent, cotangent, secant, and cosecant. Here’s why these functions are essential:

• Sine and Cosine Functions: They relate to the projections of a rotating radius on the vertical and horizontal axes, respectively, and are fundamental in modeling oscillatory phenomena like sound waves and light.
• Tangent Function: Defined as sine divided by cosine, it becomes particularly useful in determining angles of elevation or depression in real-world applications.
• Periodicity: All trigonometric functions are periodic, meaning they repeat their values in regular intervals. This property is instrumental in fields like signal processing and Fourier analysis.

When working with expressions like \( \log_{10}(\sin(x)) \), understanding the sine function's behavior within its periodic cycle is essential. This understanding helps in defining conditions under which certain logarithmic and trigonometric functions are valid and determining the nature of their graphs.