Problem 52 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to approximate the solutions of the equation in the interval \([0,2 \pi)\) by collecting all terms on one side, graphing the new equation, and using the zero or root feature to approximate the \(x\) -intercepts of the graph. $$\csc ^{2} x-3 \csc x=4$$

Step-by-Step Solution

Answer

The short answer depends on the exact approximations obtained using the graphing utility, so it cannot be given without the actual graphing process. However, it will be a list of x-values in the interval [0, 2π) where the graph touches or crosses the x-axis.

Step 1: Transforming the equation

Collect all terms on one side of the equation. The equation \(\csc ^{2} x-3 \csc x=4\) can be rewritten as \(\csc ^{2} x-3 \csc x - 4 = 0\) This form will be easier to graph since the graph of an equation y=0 will enable us to find the x-intercepts directly by using the zero or root feature of the graphing utility.

Step 2: Graphing the equation

Graph the equation \(\csc ^{2} x-3 \csc x - 4 = 0\) in the interval [0, 2π) using a graphing utility. This equation represents a curve in the xy-plane. It's important to make sure the graph is accurate in the given interval.

Step 3: Finding the x-intercepts

Observe the graph and use the zero or root feature to approximate the x-intercepts. These approximated x-values are the solutions to the equation in the given interval [0, 2π). Make sure to check both visual representation and utility's root calculation to verify these roots.

Key Concepts

Graphing Utility

A graphing utility is a powerful tool that allows you to visualize mathematical equations and functions. It makes it possible to graph complex equations easily and with precision. In this exercise, using a graphing utility helps us plot the transformed equation \[ \csc^{2} x - 3 \csc x - 4 = 0 \]By graphing this, we can visually identify where the curve crosses the x-axis within a specified interval such as \[ [0, 2\pi) \] These crossing points are crucial as they are the solutions to the equation.

• Graphing utilities can display different functions like trigonometric, exponential, and polynomial.
• They have features such as zoom, trace, and calculate, making them versatile and user-friendly.
• The zero or root function is used to estimate the x-intercepts, which are key to solving equations.

Cosecant Function

The cosecant function, denoted as \( \csc x \), is the reciprocal of the sine function. This means \[ \csc x = \frac{1}{\sin x} \] Trigonometric functions like the cosecant are key in solving problems involving periodic phenomena, oscillations, and waves. In this exercise, we manipulate the cosecant function to form the quadratic equation\[ \csc^{2} x - 3 \csc x - 4 = 0 \]

• This transformation helps simplify finding solutions within a given interval.
• As with other trigonometric functions, understanding the period and behavior of \( \csc x \) is crucial for accurate graphing.
• Remember, the \( \csc x \) function has undefined values when \( \sin x = 0 \), so these points are often skipped on the graph.

X-Intercepts

X-intercepts are points where a graph crosses the x-axis, meaning the y-value is zero. Solving the equation \[ \csc^{2} x - 3 \csc x - 4 = 0 \] involves finding these intercepts. These intercepts represent the values of \( x \) that satisfy the original equation. To accurately find the x-intercepts using a graphing utility, you can:

• Plot the equation and visually inspect where the curve hits the x-axis within the specified interval.
• Use the utility’s zero or root function to pinpoint these intersections numerically.

This method is particularly useful for complex equations where algebraic solutions are cumbersome or impossible.

Interval Notation

Interval notation provides a concise way to describe specific subsets of real numbers. In this exercise, the interval \[ [0, 2\pi) \] is used to specify the domain in which the solutions should be found. It's important to interpret this notation accurately:

• The square bracket \([0\) indicates that 0 is included in the interval.
• The round bracket at \(2\pi)\) indicates this endpoint is not included.

This notation is efficient for defining continuous ranges of numbers and is often used in calculus and algebra to indicate the domain or range of functions.