Problem 14
Question
Let \(f: R \rightarrow R\) be a function defined by \(f(x)=\\{|\cos x|\\}\), where \(\\{x\\}\) represents fractional part of \(x\). Let \(S\) be the set containing all real values \(x\) lying in the interval \([0,2 \pi]\) for which \(f(x) \neq|\cos x|\). Then, number of elements in the set \(S\) is (a) 0 (b) 1 (c) 3 (d) infinite
Step-by-Step Solution
Verified Answer
The number of elements in set \(S\) is 3.
1Step 1: Understanding the Function
The function is given by \( f(x) = \{ |\cos x| \} \), where \( \{ x \} \) denotes the fractional part of \( x \). The fractional part of a number represents its decimal part, i.e., if \( x = a + b \) where \( a \) is an integer, then \( \{ x \} = b \). Therefore, \( f(x) \) computes the fractional part of \( |\cos x| \).
2Step 2: Analyzing When the Condition Fails
We need to find when \( f(x) eq |\cos x| \), which implies \( \{ |\cos x| \} eq |\cos x| \). This happens when \( |\cos x| \) is an integer, because in such cases, the fractional part of an integer is zero, which is not equal to the integer itself unless it is 0. Hence, we focus on values where \(|\cos x|\) becomes 0 or 1, as these are the potential integer values of cosine in the interval \([0, 2\pi]\).
3Step 3: Identifying Values of x
For \(|\cos x| = 1\), \(\cos x = \pm 1\). This occurs at \(x = 0\), \(x = \pi\), and \(x = 2\pi\) for \([0, 2\pi]\). In these cases, \(f(x) = 0\) but \(|\cos x| = 1\). For \(|\cos x| = 0\), \(\cos x = 0\). This occurs at \(x = \frac{\pi}{2}\) and \(x = \frac{3\pi}{2}\). Here, \(f(x) = 0 = |\cos x|\), satisfying the condition \(f(x) = |\cos x|\).
Key Concepts
Cosine FunctionReal AnalysisTrigonometric Equations
Cosine Function
The cosine function is a fundamental trigonometric function that is crucial in various mathematical contexts. Given an angle measure, typically in radians or degrees, the cosine function calculates the adjacent side's length divided by the hypotenuse in a right triangle. The cosine function is usually denoted as \( \cos(x) \), where \( x \) represents the angle.
The values of \( \cos(x) \) range between -1 and 1 for all real numbers. This periodic function has a period of \( 2\pi \) radians, meaning that its values repeat every \( 2\pi \). Important special angles to consider are \( 0, \pi, \) and \( 2\pi \), where the cosine function reaches its maximum value of \( 1 \). Conversely, it reaches \( -1 \) at \( x = \pi \).
Additional noteworthy angles occur at \( x = \frac{\pi}{2} \) and \( x = \frac{3\pi}{2} \) where the cosine function value is zero. At these points, the graph crosses the x-axis, and the angles mark the quarter marks of a full circle on the unit circle. Understanding these special points can help when exploring the solutions of the given exercise.
The values of \( \cos(x) \) range between -1 and 1 for all real numbers. This periodic function has a period of \( 2\pi \) radians, meaning that its values repeat every \( 2\pi \). Important special angles to consider are \( 0, \pi, \) and \( 2\pi \), where the cosine function reaches its maximum value of \( 1 \). Conversely, it reaches \( -1 \) at \( x = \pi \).
Additional noteworthy angles occur at \( x = \frac{\pi}{2} \) and \( x = \frac{3\pi}{2} \) where the cosine function value is zero. At these points, the graph crosses the x-axis, and the angles mark the quarter marks of a full circle on the unit circle. Understanding these special points can help when exploring the solutions of the given exercise.
Real Analysis
Real analysis is a branch of mathematics dealing with real numbers and real-valued sequences and functions. It provides the rigorous foundation for calculus and an understanding of real functions, such as those you'd encounter in trigonometric equations.
One of the essential tools in real analysis is the concept of the fractional part function. The fractional part of a number \( x \), represented as \( \{ x \} \), is the non-integer part of \( x \). If \( x \) is expressed as \( a + b \) where \( a \) is the integer part and \( b \) is the fractional part, then \( \{ x \} = b \). This is particularly useful in scenarios involving periodic or oscillatory functions like the cosine function, where exact integer values need special attention.
Real analysis techniques are employed to explore continuity, limits, and differentiability of functions. In this context, understanding when the cosine value becomes an integer within the defined interval \([0, 2\pi]\) becomes essential. Here, \(|\cos x|\) taking on integer values simplifies to checking known key points along the unit circle.
One of the essential tools in real analysis is the concept of the fractional part function. The fractional part of a number \( x \), represented as \( \{ x \} \), is the non-integer part of \( x \). If \( x \) is expressed as \( a + b \) where \( a \) is the integer part and \( b \) is the fractional part, then \( \{ x \} = b \). This is particularly useful in scenarios involving periodic or oscillatory functions like the cosine function, where exact integer values need special attention.
Real analysis techniques are employed to explore continuity, limits, and differentiability of functions. In this context, understanding when the cosine value becomes an integer within the defined interval \([0, 2\pi]\) becomes essential. Here, \(|\cos x|\) taking on integer values simplifies to checking known key points along the unit circle.
Trigonometric Equations
Trigonometric equations involve trigonometric functions like sine, cosine, and tangent. Solving these equations often requires understanding the periodic nature of these functions and their specific values at notable angles.
An equation like \( f(x) = \{ |\cos x| \} eq |\cos x| \) highlights the need to determine when \(|\cos x|\) is an integer within \([0, 2\pi]\). This involves identifying where the cosine function hits these integer points like 0 or 1. In the case of the cosine function, this occurs at \( x = 0, \pi, 2\pi \), and special consideration is also given to \( x = \frac{\pi}{2}, \frac{3\pi}{2} \) where the cosine equals zero.
Finding solutions to trigonometric equations can therefore involve:
An equation like \( f(x) = \{ |\cos x| \} eq |\cos x| \) highlights the need to determine when \(|\cos x|\) is an integer within \([0, 2\pi]\). This involves identifying where the cosine function hits these integer points like 0 or 1. In the case of the cosine function, this occurs at \( x = 0, \pi, 2\pi \), and special consideration is also given to \( x = \frac{\pi}{2}, \frac{3\pi}{2} \) where the cosine equals zero.
Finding solutions to trigonometric equations can therefore involve:
- Identifying key angles using specific trigonometric identities or unit circle properties.
- Utilizing periodicity to predict subsequent values.
- Employing algebraic manipulation or substitution techniques when necessary.
Other exercises in this chapter
Problem 12
If \(x\) and \(\alpha\) are real, then the inequation \(\log _{2} x+\log _{x} 2+2 \cos \alpha \leq 0\) (a) has no solution (b) has exactly two solutions (c) is
View solution Problem 13
The range of values of ' \(a\) ' such that \(\left(\frac{1}{2}\right)^{|x|}=x^{2}-a\) is satisfied for maximum number of values of ' \(x\) ' (a) \((-\infty,-1)\
View solution Problem 15
The domain of the function \(f(x)=\sqrt{\log _{\sin x+\cos x}(|\cos x|+\cos x)}, 0 \leq x \leq \pi\) is (a) \((0, \pi)\) (b) \(\left(0, \frac{\pi}{2}\right)\) (
View solution Problem 16
If \(f(x)=\left(x^{2}+2 \alpha x+\alpha^{2}-1\right)^{1 / 4}\) has its domain and range such that their union is set of real numbers, then \(\alpha\) satisfies
View solution