Problem 10 - Algebra and Trigonometry

Question

Find all solutions of the equation. $$\csc ^{2} x-4=0$$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{\pi}{6} + 2k\pi, \frac{5\pi}{6} + 2k\pi, \frac{7\pi}{6} + 2k\pi, \frac{11\pi}{6} + 2k\pi\).

Step 1: Re-arrange the Equation

Begin by isolating the cosecant-squared term on one side of the equation. The given equation is \(\csc^2 x - 4 = 0\). Add 4 to both sides to obtain: \[\csc^2 x = 4\]

Step 2: Express in Terms of Sine

Recall the identity that \(\csc x = \frac{1}{\sin x}\), so \(\csc^2 x = \frac{1}{\sin^2 x}\). Substitute back into the equation to get: \[\frac{1}{\sin^2 x} = 4\]

Step 3: Solve for Sine

To solve for \(\sin^2 x\), take the reciprocal of both sides of the equation: \[\sin^2 x = \frac{1}{4}\] Then, take the square root of both sides to solve for \(\sin x\): \[\sin x = \pm \frac{1}{2}\]

Step 4: Determine the General Solutions

Based on the values obtained for \(\sin x\), we solve for \(x\) knowing the sine function values: - \(\sin x = \frac{1}{2}\) corresponds to general solutions \(x = \frac{\pi}{6} + 2k\pi\) or \(x = \frac{5\pi}{6} + 2k\pi\), where \(k\) is any integer.- \(\sin x = -\frac{1}{2}\) corresponds to general solutions \(x = \frac{7\pi}{6} + 2k\pi\) or \(x = \frac{11\pi}{6} + 2k\pi\).

Key Concepts

Cosecant Function

The cosecant function, denoted as \( \csc x \), is a trigonometric function that is essentially the reciprocal of the sine function. In simpler terms, we can express it as:

• \( \csc x = \frac{1}{\sin x} \)

The cosecant function is defined based on the sine function, making it undefined whenever the sine function is zero. This typically occurs at integer multiples of \( \pi \), where sine crosses the x-axis.

In trigonometric equations, the cosecant function often appears in squared forms, as in our exercise \( \csc^2 x - 4 = 0 \). Here, knowing the cosecant's role as the reciprocal helps in rearranging the equation to be solvable using the sine function.

The graph of \( \csc x \) shows its periodic nature and its asymptotes where the function is undefined. Recognizing these characteristics can aid in visualizing and solving trigonometric problems.

Sine Function

The sine function, expressed as \( \sin x \), is one of the fundamental trigonometric functions. It describes the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle.

• \( \sin x = \frac{\text{opposite}}{\text{hypotenuse}} \)

Sine is a periodic function with a range from -1 to 1 and a period of \( 2\pi \). This periodicity means that for values of \( x \) that differ by \( 2\pi \), the sine value will be the same.

In our exercise, the sine function was crucial when we solved \( \csc^2 x = 4 \), as it allowed us to express \( \csc^2 x \) in terms of \( \sin x \): \( \sin^2 x = \frac{1}{4} \). This simplification leads to solutions \( \sin x = \pm \frac{1}{2} \), highlighting the importance of understanding trigonometric relationships.

Knowing precise sine values for specific angles, such as \( \pi/6 \), \( 5\pi/6 \), etc., aids in finding general solutions for trigonometric equations.

Trigonometric Identities

Trigonometric identities are equations that relate different trigonometric functions to each other. They serve as crucial tools in manipulating and solving trigonometric equations. One of the key identities used in our exercise is the reciprocal identity:

• \( \csc x = \frac{1}{\sin x} \)
• \( \csc^2 x = \frac{1}{\sin^2 x} \)

This identity allows for transforming an equation involving \( \csc x \) into one involving \( \sin x \), which is often easier to solve.

Apart from reciprocal identities, there are other types crucial for various mathematical applications:

• Pythagorean identities: like \( \sin^2 x + \cos^2 x = 1 \)
• Angle sum and difference identities
• Double angle and half angle identities

In practice, these identities assist in reducing equations to simpler forms or in finding equivalent expressions that are more convenient or insightful for a particular problem.