Problem 20 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The domain of the function \(y=\sqrt{\log \frac{1}{|\sin x|}}\) is (A) \(R \backslash\\{n \pi: n \in Z\\}\) (B) \(R \backslash(-\pi, \pi)\) (C) \(R \backslash\\{2 n \pi: n \in Z\\}\) (D) \((-\infty, \infty)\)

Step-by-Step Solution

Answer

The domain is \(R \backslash \{n \pi: n \in Z\}\) (option A).

Step 1: Understanding the Function

The function given is \(y = \sqrt{\log \frac{1}{|\sin x|}}\). To find the domain, we need conditions under which both the logarithm and the square root are defined.

Step 2: Condition for Logarithm

For \(\log \frac{1}{|\sin x|}\) to be defined, \(\frac{1}{|\sin x|} > 0\). However, since \(\log\) requires positivity, we actually need \(\frac{1}{|\sin x|} > 1\), which implies \(0 < |\sin x| < 1\).

Step 3: Condition for Square Root

The expression under the square root, \(\log \frac{1}{|\sin x|}\), must be non-negative. So, \(\log \frac{1}{|\sin x|} \geq 0\), which translates to \(\frac{1}{|\sin x|} \geq 1\), meaning \(|\sin x| \leq 1\).

Step 4: Combining Conditions

Both conditions require \(0 < |\sin x| \leq 1\). However, for the logarithm to be positive, \(|\sin x|\) must be strictly between 0 and 1 but never 0 (i.e., \(\sin x eq 0\)). This occurs when \(x eq n \pi\), where \(n\) is any integer.

Step 5: Finalizing the Domain

The domain of the function is all real numbers excluding points where \(\sin x = 0\). Thus, the domain is \(R \backslash \{n \pi: n \in Z\}\), which matches option (A).

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in understanding periodic phenomena, which repeat at regular intervals. The most common trigonometric functions are sine (\( ext{sin}x\)), cosine (\( ext{cos}x\)), and tangent (\( ext{tan}x\)). Here, we focus on the sine function because it appears in the function we're analyzing.

• Sine Function: Denoted as \( ext{sin}x\), this function maps the angle \(x\) onto the range \([-1, 1]\). This indicates it can take any real number input and output values between \(-1\) and \(1\).
• Periodicity: The sine function is periodic, which means it repeats its values over regular intervals. Specifically, \( ext{sin}x\) has a period of \(2 ext{π}\). Every \(2 ext{π}\) units along the x-axis, the sine curve reproduces the same pattern.

As such, when analyzing functions involving the sine, like \(y = \sqrt{\log \frac{1}{|\sin x|}}\), it's crucial to note where the sine function is zero (\( ext{sin}x = 0\)), because that can lead to undefined values for the overall function. The values where \( ext{sin}x = 0\) occur at integer multiples of \( ext{π}\), specifically points \(n\text{\π}\), where \(n\) is any integer.

Logarithmic Functions

Logarithmic functions, denoted as \( ext{log}x\), essentially answer the question: "To what exponent must a base be raised to obtain a particular number?" For instance, in our function \(y = \sqrt{\log \frac{1}{|\sin x|}}\), we are dealing with a logarithmic expression within a square root, so understanding the properties of logarithms is pivotal.

• Definition: A function \( ext{log}b\ x\) is the inverse of the exponential function \(b^x\). It states that \(b\) raised to the \( ext{log}_b\ x\) will get back to \(x\).
• Domain of Logarithms: Logarithms are only defined for positive arguments. In other words, \(x > 0\) must hold true for a logarithm to compute a real number.

For the given function, \(\log \frac{1}{|\sin x|}\), it's recognized that the expression inside the logarithm \(\frac{1}{|\sin x|}\) must be positive and greater than one for the log to be defined as well as fit within a square root. This means that \(0 < |\sin x| < 1\) should satisfy the domain condition for the log expression, ensuring a potential domain of valid \(x\) values for the entire function.

Square Root Function

The square root function, often written as \(\sqrt{x}\), is one of the most common operations applied in mathematics, used to find a number that, when multiplied by itself, equals \(x\). When included within more complex expressions, its properties shape the conditions under which the expression is defined.

• Definition: The square root of a number \(x\) is a value \(y\) such that \(y^2 = x\). It is generally only defined for non-negative \(x\), i.e., \(x \geq 0\).
• Relevance to the Problem: In the function \(y = \sqrt{\log \frac{1}{|\sin x|}}\), this means that the expression under the square root must be non-negative. Therefore, \(\log \frac{1}{|\sin x|}\) must be greater than or equal to zero.

By these principles, the expression under the square root is only valid where the logarithm expression itself ensures \(|\sin x|\) is between zero and one, but not zero, as this would imply division by zero in the logarithmic expression. Consequently, the domain of this square root function, when put together with the logarithmic restrictions, helps determine why the sine function values cannot equal zero, finalizing the function's domain.