Problem 80 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to approximate the solutions of the equation in the interval \([\mathbf{0}, \mathbf{2} \pi)\). $$2 \tan ^{2} x+7 \tan x-15=0$$

Step-by-Step Solution

Answer

The exact solutions will vary depending upon the graphing utility used, and therefore cannot be given without the graph. Look at the x-axis intersection points to find the approximate solutions within the interval \([ 0, 2 \pi ]\).

Step 1: Rewrite the Equation

Rewrite the equation in terms of \( \tan ^ 2 x \) and \( \tan x \) like this: \( \tan ^ 2 x + \frac{7}{2} \tan x - \frac{15}{2} = 0 \) to simplify the graphing process.

Step 2: Create the Graph using a Graphing Utility

Now using a graphing utility like Desmos or Grapher, enter the above equation and create the graph. Make sure that the interval of \(x\) is \([ 0, 2 \pi ]\).

Step 3: Determine the Solutions

The solutions of the equation are the x-coordinates of the points where the graph of the function intersects the x-axis. Therefore, look at the graph and find those intersection points. Make a note of the X-coordinates of these points, as they represent the solutions of the equation in the specified interval.

Key Concepts

Graphing Utilities

Graphing utilities are powerful tools that help visualize mathematical functions and equations. They enable students and professionals alike to solve complex problems with ease. By simply inputting equations into these programs, one can quickly graph functions and observe their characteristics. This makes these tools indispensable in understanding and solving various mathematical exercises.

For our exercise, graphing utilities like Desmos or Grapher are used to visualize the quadratic equation involving the tangent function. Inputting the rewritten equation into a graphing utility allows us to see where the graph intersects the x-axis. The x-coordinates of these intersection points give us the approximate solutions to the equation.

• Visualization: Offers a visual representation, making it easier to see intersections.
• Efficiency: Saves time by quickly converting equations into graphs.
• Accuracy: Provides precise answers to mathematical problems when done correctly.

Learning how to use these utilities effectively is a crucial skill in mathematics. It simplifies the process of problem-solving, especially with complex equations.

Tangent Function

The tangent function, denoted as \( \tan x \), is a trigonometric function that is defined as the ratio of the sine function to the cosine function: \( \tan x = \frac{\sin x}{\cos x} \). It is crucial in trigonometry and appears frequently in various mathematical equations.

One key feature of the tangent function is its periodic nature, meaning it repeats at regular intervals of \( \pi \). This characteristic is particularly important in solving equations like the one in our exercise, where we seek solutions in the interval \([0, 2\pi)\).

• Periodicity: Repeats every \( \pi \), which affects the number and position of solutions.
• Asymptotes: Vertical asymptotes occur where \( \cos x = 0 \), usually at \( \frac{\pi}{2} + k\pi \).
• Range: The function spans all real numbers from \(-\infty\) to \(+\infty\).

Understanding the tangent function helps in determining the behavior of the quadratic equation in our problem, specifically how the function \( 2 \tan^2 x + 7 \tan x - 15 = 0 \) interacts with the x-axis.

Quadratic Equations

Quadratic equations are polynomial equations of the second degree, typically in the form \( ax^2 + bx + c = 0 \). They are foundational in algebra and appear in various fields, including physics, engineering, and economics.

In this exercise, the quadratic equation is expressed in terms of \( \tan x \), resulting in \( 2 \tan^2 x + 7 \tan x - 15 = 0 \). This equation can be solved either analytically or graphically by finding the values of \( \tan x \) that satisfy the equation.

• Standard Form: Written as \( ax^2 + bx + c = 0 \).
• Roots: Solutions where the equation equals zero; found using methods like factoring, using the quadratic formula, or graphing.
• Discriminant: Part of the quadratic formula \( b^2 - 4ac \) determines the nature of the roots.

In our specific problem, graphing helps approximate the solutions for \( \tan x \) within the specified interval. By treating the quadratic in terms of a single variable \( \tan x \), we can graph it and easily spot where it crosses the x-axis, giving us the solutions.