Problem 22 - A Graphical Approach to Precalculus with Limits

Question

The equation \(\cot \frac{x}{2}-\csc \frac{x}{2}-1=0\) has solution set \(\varnothing\) over the interval \([0,2 \pi) .\) The solution set to the inequality cot \(\frac{x}{2}-\csc \frac{x}{2}-1>0\) over this interval is \(\varnothing .\) Does the graph of \(y=\cot \frac{x}{2}-\csc \frac{x}{2}-1\) lie above or below the \(x\) -axis over this interval?

Step-by-Step Solution

Answer

The graph lies below the x-axis over the interval \([0,2\pi)\).

Step 1: Analyze the Equation

The given equation is \( \cot \frac{x}{2} - \csc \frac{x}{2} - 1 = 0 \). Substituting the trigonometric identities, we have \(\cot \frac{x}{2} = \frac{\cos \frac{x}{2}}{\sin \frac{x}{2}}\) and \(\csc \frac{x}{2} = \frac{1}{\sin \frac{x}{2}}\). Therefore, the equation becomes \( \frac{\cos \frac{x}{2}}{\sin \frac{x}{2}} - \frac{1}{\sin \frac{x}{2}} - 1 = 0 \). This simplifies to \( \cos \frac{x}{2} - 1 = \sin \frac{x}{2} \).

Step 2: Simplify the Equation

Rearranging gives \( \cos \frac{x}{2} - \sin \frac{x}{2} = 1 \). Squaring both sides results in \(\cos^2 \frac{x}{2} - 2\cos \frac{x}{2} \sin \frac{x}{2} + \sin^2 \frac{x}{2} = 1 \). Using the Pythagorean identity \(\cos^2 \frac{x}{2} + \sin^2 \frac{x}{2} = 1\), this reduces to \(-2\cos \frac{x}{2} \sin \frac{x}{2} = 0\). This implies \(\sin x = 0\).

Step 3: Solve for x

The solutions to \(\sin x = 0\) in the interval \([0, 2\pi)\) are \(x = 0, \pi\). However, these solutions lead to undefined values in the original equation due to division by zero, thus there is actually no solution: \(\varnothing\).

Step 4: Analyze the Inequality

Now consider the inequality \(\cot \frac{x}{2} - \csc \frac{x}{2} - 1 > 0\). If the same trigonometric substitutions are made, we have \( \cos \frac{x}{2} - \sin \frac{x}{2} > 1 \), and following steps, results in \(-2\cos \frac{x}{2} \sin \frac{x}{2} > 0\) which cannot be satisfied for any \(x\) in \([0, 2\pi)\).

Step 5: Determine the Graph Position

Since both the equation and inequality resulted in \(\varnothing\), the function \(y = \cot \frac{x}{2} - \csc \frac{x}{2} - 1\) does not cross the x-axis, nor does it ever take positive values within this interval. Thus, it must lie entirely below the x-axis on \([0, 2\pi)\).

Key Concepts

Cotangent Function

The cotangent function, often abbreviated as \( \cot \theta \), is one of the primary trigonometric functions. It is the reciprocal of the tangent function. The definition of the cotangent function can be written as \( \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \). This expression shows that the cotangent of an angle is the ratio of the adjacent side to the opposite side in a right triangle.In the given problem, the expression \( \cot \frac{x}{2} \) is utilized. This represents the cotangent of half the angle \( x \). Using identities like this is often necessary to simplify and solve equations and inequalities involving trigonometric functions. This function is undefined when \( \sin \frac{x}{2} = 0 \), as division by zero is not possible, leading to gaps or undefined values in its graph.

Cosecant Function

Cosecant function, notated as \( \csc \theta \), is another critical trigonometric function that represents the reciprocal of the sine function. It can be defined as \( \csc \theta = \frac{1}{\sin \theta} \). This function is vital in calculations where angles and ratios of a right triangle's sides are involved.In our exercise, we encounter \( \csc \frac{x}{2} \), indicating the cosecant function applied to half of the angle \( x \). It's crucial to remember that \( \csc \theta \) is undefined when \( \sin \theta = 0 \), since division by zero results in undefined expressions. This detail is critical as it influences the domain of the functions and their graphs.The applicability of the cosecant function shares many attributes with other trigonometric functions, but its undefined nature at specific points often plays a role in solution sets, as seen in our problem where no valid solutions were found.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all angles for which the functions are defined. These identities are essential tools in simplifying complex trigonometric equations and proving other mathematical assertions.For instance, some common identities include:

• Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \).
• Reciprocal identities: \( \csc \theta = \frac{1}{\sin \theta} \), \( \sec \theta = \frac{1}{\cos \theta} \), and \( \cot \theta = \frac{1}{\tan \theta} \).
• Quotient identities: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), and \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).

These identities are pivotal in understanding and solving the problem where substituting \( \cot \frac{x}{2} \) and \( \csc \frac{x}{2} \) facilitated the simplification of the original equation. Such substitutions help transform complex expressions into more manageable forms, enabling easier problem-solving.

Inequalities in Trigonometry

Inequalities involving trigonometric functions often require a mix of identities and functionality of these functions to find a solution or to prove the inequality holds. They present a way to determine where a function may be greater than or less than a certain value across an interval.In the given problem, the inequality \( \cot \frac{x}{2} - \csc \frac{x}{2} - 1 > 0 \) centers around using identities and logical reasoning to determine under what conditions this expression holds true. However, through algebraic manipulation and with understanding the range and behavior of these functions, it was found there are no solutions in the interval \([0, 2\pi) \).When solving such inequalities, always watch for the domains of involved functions, as they may not be defined for all values (such as division by zero instances). Properly applying algebraic steps and keeping an eye out for undefined regions or behaviors helps conclude if solutions exist, or if an inequality holds strong over a specific interval.