The Graph of y Log(Sin(x)): An In-depth Analysis

The logarithm of the sine function, denoted as y log(Sin(x)), is a fascinating topic in mathematics. This article delves into the visualization, behavior, and key properties of this function. We will explore the graph, asymptotes, and periodicity of y log(Sin(x)) using R software and mathematical reasoning.

Understanding the Function Domain

The function log(Sin(x)) is only valid for positive values of Sin(x). This implies that Sin(x) must lie in the interval . Therefore, the domain of x is , and the function has a periodicity of 2π, due to the periodic nature of Sin(x).

Graphing y Log(Sin(x)) Using R

To visualize the graph of y log(Sin(x)), we can use the R software. Below is the R code to graph this function:

plot(x, log(sin(x)), type 'l', main 'Plot of y Log(Sin(x))', xlab 'x', ylab 'Log(Sin(x))')

Running this code will generate a plot that clearly illustrates the behavior of y log(Sin(x)). The graph will be negative for all values within the domain of the function, with symmetry around x π/2.

Asymptotes and Symmetry

The graph of y log(Sin(x)) has vertical asymptotes at integral multiples of π. This is because log(Sin(x)) is undefined at these points, where Sin(x) 0. These asymptotes create a visual indication of where the function tends towards negative infinity as x approaches integer multiples of π.

The function y log(Sin(x)) is symmetric about x π/2. This symmetry is a result of the symmetric nature of the Sin(x) function around this point. For x π/2, the function value is 0, as Sin(π/2) 1 and log(1) 0.

Key Properties and Critical Points

Let's define the function f(x) Sin(x) * Log(Sin(x)). Analyzing the critical points and behavior of f(x) provides insight into the function's shape:

By setting f(x) Sin(x) * Log(Sin(x)), we notice that f(π - x) f(x). This shows that the function is symmetric about x π/2. The function f(x) is always non-positive, that is, f(x) ≤ 0 for all x in the domain 0 As x approaches 0 or π, the function tends to zero due to the indeterminate form x * log(x) which is proven to be 0 using elementary calculus limits.

The first derivative of f(x) is given by:

f'(x) Cos(x) * Log(Sin(x)) Cos(x) * (-1/Sin(x))

A critical point occurs when the first derivative is zero. This happens at x π/2, and at two points where Log(Sin(x)) -1, which yields x arcsin(1/e) and x π - arcsin(1/e).

The second derivative helps in understanding the concavity and inflexion points of the function:

f''(x) -Sin(x) * Log(Sin(x)) (Cos^2(x)/Sin(x)) - Sin(x)

By studying the function g(t) 2 - 4t - t * log(t) for 0

Conclusion

In conclusion, the graph of y log(Sin(x)) demonstrates a complex interplay between the logarithmic and trigonometric functions, with vertical asymptotes, periodicity, and unique critical points. Understanding these properties not only helps in analyzing the function but also provides insights into the behavior of similar functions in advanced mathematics.