Problem 52 - Precalculus with Limits

Question

In Exercises 49-58, use a graphing utility to approximate the solutions (to three decimal places) of the equation in the interval \( [0, 2\pi) \). \( \dfrac{\cos x \cot x}{1 - \sin x} = 3 \)

Step-by-Step Solution

Answer

The exact solutions will depend on the result graphed on the graphing utility. Record the x-coordinate of the points at which the graph intersects the x-axis. These are the approximate solutions to the given equation within the interval \( [0, 2\pi) \).

Step 1: Rewrite the Equation

First, rewrite the equation to one side. Simplify it to: \( \dfrac{\cos x \cot x}{1 - \sin x} - 3 =0 \)

Step 2: Graph the Equation

Use a graphing utility to graph the equation. Make sure that the range of the graph includes the interval \( [0,2\pi) \). Observe the intersect with the x-axis.

Step 3: Find the Solutions

Identify the points at which this graph intersects the x-axis. These will be the solutions of the equation. Record the x-coordinates of these points to three decimal places.

Key Concepts

Graphing Calculator

A graphing calculator is an essential tool when dealing with complex equations like trigonometric functions. It allows you to visualize the equation and find solutions more efficiently. Using a graphing calculator, you can:

• Plot the graph of the mathematical function accurately
• Identify key graphical features such as intercepts and points of intersection
• Zoom in and out for a better view of certain parts of the graph

In the context of the problem given, the graphing calculator is used to plot the equation \[ \frac{\cos x \cot x}{1 - \sin x} - 3 = 0 \]on the interval \([0, 2\pi)\). By visualizing this on the calculator, you can easily identify the points where the graph intersects the x-axis. These points correspond to the solutions of the original equation. This tool is invaluable for solving, plotting, and analyzing mathematical equations effectively and is commonly used in educational settings.

Trigonometric Functions

Trigonometric functions are the backbone of trigonometry, and they are heavily utilized in many mathematical problems. In the given exercise, we encounter the cosine \( \cos x \) and cotangent \( \cot x \) functions.

• The cosine function, denoted as \( \cos x \), represents the x-coordinate of a point on the unit circle corresponding to an angle x in radians.
• The cotangent function, denoted as \( \cot x \), is the reciprocal of the tangent \( (\tan x) \), which can also be expressed as \( \frac{\cos x}{\sin x} \).

These functions are periodic, meaning they repeat their values in regular intervals over their domains. The period of \( \cos x \) is \( 2\pi \), while for \( \cot x \), it is \( \pi \). Understanding these periods is essential when solving equations over specified intervals, such as \([0, 2\pi)\).

Interval Notation

Interval notation is a way of representing a range of values. In mathematics, it is particularly useful for describing the domain and range of functions or the solution sets for equations. The interval \([0, 2\pi)\) provided in the exercise specifies that solutions should be found between 0 and \( 2\pi \), including 0 but not \( 2\pi \).

• A square bracket \([\ ]\) indicates that the endpoint is included in the interval, known as closed interval.
• A parenthesis \((\ \)) signifies that the endpoint is not included, known as open interval.

This notation is compact and avoids unnecessary verbosity, making it clear what range of values is considered. It is essential to understand how to read and write interval notation, especially when dealing with periodic functions and graphing calculators in mathematics.