\frac{\sin x \ln (\cos x)}{\cos x}
Answer (2)
The given expression is c o s x s i n x l n ( c o s x ) .
The domain of the expression is determined by the condition 0"> cos x > 0 , which means x must be in the interval ( − 2 π + 2 πk , 2 π + 2 πk ) for any integer k .
The expression can be rewritten as tan x ln ( cos x ) .
As x approaches the boundaries of its domain, the expression approaches an indeterminate form of the type ∞ ⋅ ( − ∞ ) .
Explanation
Problem Analysis We are given the expression c o s x s i n x l n ( c o s x ) . Our goal is to analyze this expression, which includes determining its domain, rewriting it in a simplified form, and understanding its behavior.
Determining the Domain First, let's determine the domain of the expression. The expression involves ln ( cos x ) , which requires that 0"> cos x > 0 . Also, since cos x is in the denominator, we must have cos x = 0 . Combining these two conditions, we require 0"> cos x > 0 . This occurs when x lies in the interval ( − 2 π + 2 πk , 2 π + 2 πk ) for any integer k .
Rewriting the Expression Next, we can rewrite the expression using trigonometric identities. We know that tan x = c o s x s i n x . Therefore, we can rewrite the given expression as tan x ln ( cos x ) .
Analyzing the Behavior Now, let's analyze the behavior of the expression. As x approaches − 2 π or 2 π from within its domain, cos x approaches 0. Thus, ln ( cos x ) approaches − ∞ . At the same time, tan x approaches − ∞ as x approaches − 2 π and tan x approaches ∞ as x approaches 2 π . Therefore, we have an indeterminate form of the type ∞ ⋅ ( − ∞ ) or ( − ∞ ) ⋅ ( − ∞ ) .
Examples
Understanding the behavior of trigonometric and logarithmic functions is crucial in many areas of physics and engineering. For example, in signal processing, analyzing the domain and behavior of such functions helps in designing filters and understanding the stability of systems. Similarly, in optics, the intensity of light can be modeled using trigonometric functions, and understanding their logarithmic relationships helps in analyzing light attenuation.
The expression c o s x s i n x l n ( c o s x ) is analyzed by determining its domain, which is affected by 0"> cos x > 0 . It can be rewritten as tan x ln ( cos x ) , and approaches an indeterminate form as x approaches its domain boundaries. Understanding this behavior is crucial in analyzing limits in calculus.
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