Problem 39 - 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. $$\csc ^{2} x-2=0$$

Step-by-Step Solution

Answer

The solutions to the equation \(\csc ^{2} x-2=0\) in the interval \([0,2 \pi)\) are \(x = \frac{\pi}{4}\), \(\frac{3\pi}{4}\), \(\frac{5\pi}{4}\), and \(\frac{7\pi}{4}\)

Step 1: Rewrite in terms of Sine

Rewrite the equation \(\csc ^{2} x-2=0\) in terms of the sine function. Recall that \(\csc x = \frac{1}{\sin x}\). So, the equation can be rewritten as \((\frac{1}{\sin x})^2 - 2 = 0\)

Step 2: Simplify the Equation

Next, simplify the equation so that it looks more recognizable and workable. We have \(\frac{1}{\sin^{2} x}=2\), or equivalently, \(\sin^{2} x=\frac{1}{2}\)

Step 3: Solve for Sine

Next, solve the simplified equation for \(\sin x\), we will get \(\sin x = \pm \sqrt{\frac{1}{2}}\) or \(\sin x = \pm\frac{1}{\sqrt{2}}\)

Step 4: Identify the solutions in the desired interval

The solutions of \(\sin x = \pm\frac{1}{\sqrt{2}}\) are x= \(\pm \frac{\pi}{4}\), \( \pm \frac{3\pi}{4}\), \( \pm \frac{5\pi}{4}\), and \( \pm \frac{7\pi}{4}\). However, given the interval \([0, 2\pi)\), we only consider the positive solutions. So the solutions to the equation in the interval are x= \(\frac{\pi}{4}\), \( \frac{3\pi}{4}\), \( \frac{5\pi}{4}\), and \( \frac{7\pi}{4}\)

Step 5: Verify Numerically

Lastly, to verify our solutions, use the table feature of a graphing utility to check the values of x where the equation \(\csc ^{2} x-2=0\) holds true. The values should align with our findings in step 4

Key Concepts

Understanding the Cosecant Function

The cosecant function, often denoted as \( \csc x \), is the reciprocal of the sine function. If you know \( \sin x \), then \( \csc x = \frac{1}{\sin x} \). This relationship is crucial for transforming equations involving trigonometric functions. In the problem you've encountered, rewriting the equation \( \csc^2 x - 2 = 0 \) in terms of sine is the first step. By understanding the reciprocal nature of cosecant, you can switch easily between these functions. This allows for simpler problem-solving and more familiar algebraic forms.

Finding Interval Solutions

Solving trigonometric equations within a specified interval requires careful attention to the range you're working within. In this case, the interval \([0, 2\pi)\) is key. After rewriting the equation in terms of \( \sin x \), you solve for values that satisfy \( \sin^2 x = \frac{1}{2} \). This gives solutions like \( \sin x = \pm \frac{1}{\sqrt{2}} \) within the interval. It's essential to focus only on those angles between 0 and \( 2\pi \) for this type of problem. This step ensures you provide all valid solutions relevant to the problem's constraints.

Connecting to the Sine Function

The sine function is a fundamental aspect of trigonometric equations. In this scenario, by rewriting the cosecant equation as one involving sine, \( \frac{1}{\sin^2 x} = 2 \), you simplify the problem significantly. This leads to \( \sin^2 x = \frac{1}{2} \), which further reduces to \( \sin x = \pm \frac{1}{\sqrt{2}} \). Recognizing the connection between these forms aids in identifying solutions within the specified interval. Familiarity with the unit circle can also help, as angles like \( \frac{\pi}{4} \) naturally arise from common trigonometric values.

Using a Graphing Utility

Graphing utilities, such as graphing calculators or software, are excellent tools for verifying solutions. With the table feature, you can input values and check whether they satisfy the original equation. In practice, you verify that \( \csc^2 x - 2 = 0 \) holds true for the solutions you found, such as \( \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \) and \( \frac{7\pi}{4} \). This step assures you that your mathematical reasoning is correct, providing a visual or numerical affirmation of your algebraic solutions.