Problem 8 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(e^{-\pi 2}<\theta<\pi / 2\), then (A) \(\cos \log \theta<\log \cos \theta\) (B) \(\cos \log \theta>\log \cos \theta\) (C) \(\cos \log \theta \leq \log \cos \theta\) (D) none of these

Step-by-Step Solution

Answer

(B) \(\cos \log \theta > \log \cos \theta\)

Step 1: Analyze the Range

We know that \(e^{-rac{ heta}{2}}<\theta<\frac{\pi}{2}\). Find \(\log(\theta)\) to determine the behavior of the \(\cos\) and \(\log\) functions in this range.

Step 2: Evaluate \(\log(\theta)\)

Given the range of \(\theta\), compute \(\log(\theta)\). For \(\theta>e^{-\frac{\pi}{2}}\), we need to ensure that \(\log(\theta)\) is negative because the logarithm is decreasing when \(\theta\) is less than 1.

Step 3: Compute \(\cos(\log(\theta))\)

Under the assumption that \(\log(\theta) < 0\), \(\cos(\log(\theta))\) will evaluate to a value close to 1 because cosine of negative numbers is positive (examined around angles close to 0).

Step 4: Evaluate \(\log(\cos(\theta))\)

In the given range \(\theta<\frac{\pi}{2}\), \(\cos(\theta)\) is decreasing, being less than 1. Thus, \(\log(\cos(\theta))\) is negative. Calculate using approximate values.

Step 5: Compare Values

Compare the results of \(\cos(\log(\theta))\) and \(\log(\cos(\theta))\). In this scenario, \(\cos(\log(\theta))\) is close to 1, hence positive, whereas \(\log(\cos(\theta))\) is negative.

Key Concepts

Logarithmic Inequalities

Understanding logarithmic inequalities helps solve many mathematical problems, including the one at hand. Logarithmic functions, indicated as \(\log(x)\), describe an inverse relationship to exponential functions. They often deal with numbers less than one, where the logarithm gives negative values. In the given problem, the variable \(\theta\) falls between \(e^{-\frac{\pi}{2}}\) and \(\frac{\pi}{2}\). It is crucial to compute \(\log(\theta)\) within this range to analyze the inequalities properly.

• The property of logarithms is such that for \(0<\theta<1\), \(\log(\theta)\) is always negative.
• This negativity aspect influences the ways inequalities are set up and compared: smaller inputs result in more negative values.

Knowing this behavior simplifies the problem, as once \(\log(\theta)\) is known, it guides the understanding of connected functions like the cosine in exponential expressions.

Cosine Function

The cosine function, represented as \(\cos(x)\), is a trigonometric function particularly helpful when analyzing angular relationships in various applied contexts. In mathematical expressions, the cosine function is periodic, with a cycle every \(2\pi\) radians, and it mitigates between values -1 and 1. This characteristic enables it to evaluate certain expressions effectively, such as during cosine-logarithmic comparisons.In the given exercise,

• The expression \(\cos(\log(\theta))\) is calculated by examining the logarithm of \(\theta\).
• Given that the value of \(\log(\theta)\) is negative, the cosine will yield a result that is positive.
• Cosine values tend to approach 1 when examined around angles near zero.

Thus, when \(\log(\theta)\) indicates small negative values, \(\cos(\log(\theta))\), naturally approaches unity.

Range Analysis

Understanding range analysis is critical when computing and comparing expressions involving functions like cosine and logarithms. It involves evaluating limits within expressions to better understand how the functions interact.In our exercise, the analysis starts by determining the permissible values of \(\theta\) which are between \(e^{-\frac{\pi}{2}}\) and \(\frac{\pi}{2}\).

• This range ensures \(\theta\) is positive and less than \(\pi/2\).
• When \(\theta<\pi/2\), it guarantees that \(\cos(\theta)\), the output is positive yet less than one.
• Interpreting \(\log(\cos(\theta))\), it results in a negative value due to the smaller output of \(\cos(\theta)\) (being less than one), while confirming logarithm properties.

This range examination allows pairing of outcomes for a proper contrast with the cosine of the logarithmic value, elucidating the provided trigonometric inequality.