Problem 53 - 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. $$\csc x+\cot x=1$$

Step-by-Step Solution

Answer

The answer is approximating visually through graphing the equation in graphing utility. The exact root values will be the x-values where the graph intersects with the x-axis, which vary slightly depending on the accuracy of the graphing utility.

Step 1: Manipulation of the Equation

The given equation is: \(\csc x+\cot x=1\). To solve for this equation, we have to rewrite this equation into a familiar form so that we can graph it using a graphing utility. As \(\csc x\) is equivalent to \(1/\sin x\) and \(\cot x\) to \(1/\tan x\) which can be written as \(\cos x / \sin x\), the equation becomes: \(1/\sin x + \cos x / \sin x = 1\). By multiplying by \(\sin x\) throughout, we get \(1+\cos x = \sin x\). Therefore, the simplified form of the equation is \(\sin x - \cos x - 1=0\).

Step 2: Graphing the Equation

We graph the equation \(\sin x - \cos x - 1=0\) on the interval from 0 to \(2 \pi\). Using a graphing utility, we see the curve where it intersects with the x-axis.

Step 3: Approximating the Solutions

The solutions of the equation will be the x-values where the graph intersects with the x-axis on the interval from 0 to \(2 \pi\). We use the zero or root feature of our graphing utility to find these points. Note that the intersect at x-axis are the solutions.

Key Concepts

Graphing Utilities

Graphing utilities are tools that help visualize mathematical functions. These tools allow students to see the graphical representation of equations, making it easier to understand where potential solutions (like x-intercepts) exist. By inputting an equation into a graphing utility, you can see the curve formed by the equation over a specified interval.

To find the solutions of an equation using a graphing utility, you can collect all terms on one side of the equation and graph the resulting expression. In this case, we're looking for x-values where the graph crosses the x-axis, representing solutions or "roots" of the equation.

• **Inputting the Equation**: First, enter the rewritten equation: \( \sin x - \cos x - 1 = 0 \) into the graphing utility.
• **Using Tools**: Most utilities have features to find intersections or zeros. Use these to locate where the graph touches the x-axis.
• **Interval**: Ensure you're only looking at the interval from 0 to \( 2 \pi \) as specified by the problem.

Graphing utilities streamline solving trigonometric equations by providing a visual approach to identifying solutions.

Cosecant Function

The cosecant function, denoted as \( \csc x \), is the reciprocal of the sine function. It is defined as: \[ \csc x = \frac{1}{\sin x} \]\This means it is undefined wherever \( \sin x = 0 \) because division by zero is undefined.

Understanding the cosecant function's behavior is crucial when dealing with its corresponding equations. It has some key characteristics:

• **Periodicity**: Like other trig functions, \( \csc x \) is periodic, with a period of \( 2\pi \).
• **Asymptotes**: It has vertical asymptotes wherever the sine function is zero (e.g., \( x = 0, \pi, 2\pi \)).
• **Graph Shape**: Its graph consists of alternating U-shaped sections above and below the x-axis where \( \sin x eq 0 \).

In this exercise, understanding \( \csc x \) helps us convert \( \csc x + \cot x = 1 \) to a format more easily graphed. Recognizing these properties assists in analyzing and solving trigonometric equations.

Cotangent Function

The cotangent function, \( \cot x \), is another fundamental trigonometric function. It is defined as the reciprocal of the tangent function: \[ \cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x} \] This formula is useful when rewriting equations involving cotangent.

Key points about the cotangent function include:

• **Periodicity**: The cotangent function is periodic with a period of \( \pi \).
• **Asymptotes**: It has vertical asymptotes at points where \( \tan x = 0 \) (e.g., \( x = 0, \pi, 2\pi, \dots \)).
• **Graph Shape**: Its graph features downward sloping lines over each interval, interspaced with asymptotes.

The cotangent's behavior is particularly important in transforming the equation for graphing. By expressing \( \cot x \) as \( \frac{\cos x}{\sin x} \), it can be seamlessly integrated with the cosecant function to form a single trigonometric expression amenable to graphing. Understanding these details helps ensure accurate graphing and solution approximation.