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Chapter 5: Problem 10

Verifying Solutions In Exercises \(5-10\) , verify that the \(x\) -values are solutions of the equation. $$\begin{array}{l}{\csc ^{4} x-4 \csc ^{2} x=0} \\ {\text { (a) } x=\frac{\pi}{6}}\end{array}$$ $$x=\frac{5 \pi}{6}$$

Short Answer

Expert verified
Both \(x = \frac{\pi}{6}\) and \(x = \frac{5\pi}{6}\) are solutions to the given equation.

Step by step solution

01

- Substitute first \(x\)-value into the equation

Replace \(x\) with \(\frac{\pi}{6}\) in the given equation, we obtain: \(\csc ^{4}(\frac{\pi}{6})-4 \csc ^{2} (\frac{\pi}{6}) = 0\). Now calculate the values of \(\csc ^{4}( \frac{\pi}{6})\) and \(\csc ^{2}( \frac{\pi}{6})\)
02

- Calculate and simplify

We know that \(\csc(\frac{\pi}{6}) = \frac{1}{\sin(\frac{\pi}{6})} = 2\). Thus, the equation becomes: \((2)^{4}-4* (2)^{2}= 0\), which simplifies to \(16-16 = 0\). Thus, for \(x = \frac{\pi}{6}\), the equation holds, making it a solution to the equation.
03

- Substitute second \(x\)-value into the equation

Replace \(x\) with \(\frac{5\pi}{6}\) in the equation, and we get: \(\csc ^{4}(\frac{5\pi}{6})-4 \csc ^{2} (\frac{5\pi}{6}) = 0\). Now calculate the values of \(\csc ^{4}(\frac{5\pi}{6})\) and \(\csc ^{2}(\frac{5\pi}{6})\).
04

- Calculate and simplify

We know that \(\csc(\frac{5\pi}{6}) = \frac{1}{\sin(\frac{5\pi}{6})} = 2\). Thus, the equation becomes: \((2)^{4}-4* (2)^{2} = 0\), which simplifies to \(16-16 = 0\). Thus, for \(x = \frac{5\pi}{6}\), the equation holds, making it a solution to the equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cosecant Function
The cosecant function, denoted by \( \text{csc}(x) \), is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the sine function, so \( \text{csc}(x) = 1/ \text{sin}(x) \) whenever \( \text{sin}(x) \) is not equal to zero.

To differentiate the use of the cosecant function in equations, it's important to remember that it exhibits particular values at specific angles. For example, \( \text{csc}(\frac{\pi}{6}) = 2 \), because \( \text{sin}(\frac{\pi}{6}) = \frac{1}{2} \). This reciprocal relationship is pivotal when verifying solutions for trigonometric equations involving the cosecant function.

When dealing with trigonometric functions, understanding the unit circle and common trigonometric values is essential. Being familiar with these can significantly simplify calculations and problem-solving. It's imperative to know these well-established values, as they're frequently used in a variety of mathematical contexts.
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variable for which the function is defined. They are used to simplify trigonometric expressions, and to prove other mathematical concepts.

Common examples include the Pythagorean identities, such as \( \text{sin}^2(x) + \text{cos}^2(x) = 1 \), and the angle sum and difference identities, for instance, \( \text{sin}(x \text{\textpm} y) = \text{sin}(x) \text{cos}(y) \text{\textpm} \text{cos}(x) \text{sin}(y) \). These identities are key in solving trigonometric equations as they allow for the transformation of one trigonometric form into another, hence simplifying complex expressions.

Understanding and memorizing these identities enable students to navigate trigonometric problems more easily, particularly in verifying trigonometric solutions, where the identities can make or break the simplification process.
Solving Trigonometric Equations
Solving trigonometric equations is an exercise in applying trigonometric identities and known values of trigonometric functions at specific angles to find the values of the variable(s) that satisfy the equation.

When attempting to solve trigonometric equations, it is beneficial to follow a structured approach:
  • Firstly, simplify the equation using known identities.
  • Secondly, substitute known values of trigonometric functions (such as \( \text{sin}(\frac{\pi}{6}) \) or \( \text{csc}(\frac{\text{pi}}{6}) \)).
  • Lastly, solve for the variable.
This methodical approach can also be used for verifying potential solutions by plugging them into the original equation and applying simplification techniques to assess their validity.

Equipped with the knowledge of trigonometric identities and the properties of trigonometric functions, it becomes more manageable to tackle trigonometric equations. Continuous practice is essential for gaining proficiency in this area, which, in essence, can bolster a student's overall mathematical skill set.

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