Problem 45 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find all solutions of the equation in the interval \([0,2 \pi)\) algebraically. Use the table feature of a graphing utility to check your answers numerically. $$2 \sin x+\csc x=0$$

Step-by-Step Solution

Answer

The equation has no real solutions in the interval [0, 2π).

Step 1: Rewrite the equation

Rewrite \(\csc x\) as \(1/\sin x\). Thus, the equation becomes \(2 \sin x + \frac{1}{\sin x} = 0\).

Step 2: Convert to Quadratic form

Multiply each term in the equation by \(\sin x\) to eliminate the fraction. We have \(2 \sin^2 x + 1 = 0\).

Step 3: Re-arrange the equation

Rearrange the quadratic equation to look like a standard quadratic equation. The equation becomes \(2 \sin^2 x = -1\).

Step 4: Solve for \(\sin x\)

Isolate \(\sin x\) on one side of the equation to calculate the value of \(\sin x\). Here we find that \(\sin x = \sqrt{-1/2}\).

Step 5: Recognize that there's no real solutions

Notice that the square root of a negative number doesn't have real solutions. This means there are no real solutions to this equation in the interval [0, 2π).

Key Concepts

Trigonometric Identities

Trigonometric identities are algebraic equations involving trigonometric functions that are true for every value of the occurring variables. They are essential tools in simplifying and solving trigonometric equations.
In the given problem, we encounter the identity that relates sine and cosecant:

• The cosecant function, \( ext{csc} \, x\), is the reciprocal of the sine function, \( ext{sin} \, x\).

This identity is critical as it helps us rewrite the equation in terms of a single trigonometric function. By converting \( ext{csc} \, x\) to \(rac{1}{ ext{sin} \, x}\), we simplify the equation and make it easier to manipulate algebraically. Often, transforming all functions to like terms allows us to apply additional algebraic techniques.
Recognizing and effectively using trigonometric identities can simplify complex expressions and aid in ensuring solutions are within desired parameters. Understanding these identities is vital for mastering trigonometry problems.

Quadratic Equations

Quadratic equations are polynomial equations of the form \(ax^2 + bx + c = 0\), where \(a, b, \) and \(c\) are constants and \(a eq 0\). They play a central role in algebra and often appear in various mathematical contexts, including trigonometry.
In the given problem, after rewriting and simplifying the trigonometric equation, it is transformed into a quadratic form: \(2\sin^2\,x + 1 = 0\). This transformation is key because:

• It allows us to apply quadratic solving techniques, such as factoring, completing the square, or using the quadratic formula.
• It provides a strategic approach to determine the existence of solutions.

The notable aspect here is that the equation \(2\sin^2\,x = -1\) indicates that \(\sin^2\,x\) must be negative, which is not possible because sine values in this context range from \(-1 \) to \(1\). Recognizing these boundaries helps determine that there are no real solutions, highlighting how quadratic forms and their intrinsic properties guide us in problem-solving.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. It is especially useful for expressing the domain or range of a function or determining valid solutions within a specific scope. In this exercise, we use the interval \([0, 2\pi)\), indicating:

• The solutions must be in the realm from \(0\) inclusive to \(2\pi\) exclusive.

Using interval notation is crucial because:

• It specifies the precise boundaries within which any solution must fall, ensuring clarity and understanding of the solution's context.
• It assists in comprehensively checking for solutions that fit the criteria imposed by the problem.

In our case, verifying that no values of \(x\) within this interval satisfy the transformed equation signifies that the trigonometric equation has no real solutions in the given domain. Utilizing interval notation efficiently guides both the solving process and verification of possible results.