Problem 61 - Precalculus with Limits

Question

In Exercises 59-62, use the Quadratic Formula to solve the equation in the interval \( [0, 2\pi) \). Then use a graphing utility to approximate the angle \( x \). \( \tan^2 x + 3 \tan x + 1 = 0 \)

Step-by-Step Solution

Answer

The solutions to the quadratic equation will be the two values of \( \tan x \). The corresponding angles can be found using Arctangent or an equivalent function, ensuring they lie within the interval [0, \(2\pi\)). These angles will be confirmed using a graphing utility.

Step 1: Write the equation in standard form

The given equation is already in the standard quadratic form, \( ax^2 + bx + c = 0 \), where a = 1, b = 3, and c = 1.

Step 2: Use the Quadratic Formula

The Quadratic Formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), can be used to solve for \( \tan x \). The substituted values are \( a = 1, b = 3, c = 1 \). This will give us two possible values for \( \tan x \).

Step 3: Solve for \( \tan x \)

After substituting the values for a, b, and c into the Quadratic Formula, solve for \( \tan x \). This results in two values for \( \tan x \).

Step 4: Solve for x

Once we have the values for \( \tan x \), we can find the angle x by using the Arctangent function or a function similar to it. Ensure that the result lies in the interval [0, \(2\pi\)). If it doesn't, adjust the result by adding or subtracting \(2\pi\), if necessary.

Step 5: Check the result with a graphing utility

Finally, a graphing utility can be utilized to approximate the angle x, and confirm the derived solution.

Key Concepts

Tan Function

The tangent function, often represented as \( \tan(x) \), is a fundamental trigonometric function. It is the ratio of the sine and cosine functions: \( \tan(x) = \frac{\sin(x)}{\cos(x)} \). This means the tangent function will be undefined whenever \( \cos(x) = 0 \), leading to vertical asymptotes and periodic interruptions in its graph.

Key characteristics of the tangent function include:

• Periodicity of \( \pi \), meaning that \( \tan(x + \pi) = \tan(x) \) for all \( x \).
• Zeroes occurring at integer multiples of \( \pi \), such as \( 0, \pi, 2\pi, \) etc.
• Vertical asymptotes at odd multiples of \( \frac{\pi}{2} \).

Understanding these properties helps us analyze and solve problems involving the tangent function, especially when finding solutions in trigonometric equations or working with intervals such as \( [0, 2\pi) \). This can also guide the determination of additional solutions that lie outside the principal range by adding or subtracting multiples of \( \pi \).

Standard Form Equation

A standard form equation refers to a specific arrangement of terms. In algebra, this often means organizing the equation as \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) are constants, and the equation is set equal to zero.

In trigonometry, when dealing with equations like the one in our exercise, the focus shifts slightly, but the concept of forming a standard quadratic equation remains.

• Identify the coefficients: For \( \tan^2 x + 3 \tan x + 1 = 0 \), \( a = 1 \), \( b = 3 \), and \( c = 1 \).
• Use these coefficients to employ solutions algorithms such as the Quadratic Formula.

This approach allows us to utilize algebraic techniques to solve trigonometric equations, seamlessly integrating these two branches of mathematics. Simplifying complex trigonometric problems using the standard form is especially helpful when applying methods such as the Quadratic Formula to find unknown values.

Trigonometric Equations

Trigonometric equations involve trigonometric functions like sine, cosine, and tangent. Solutions to these equations often require understanding both algebraic and trigonometric principles.

The quadratic trigonometric equation given in the exercise is an example: \( \tan^2 x + 3 \tan x + 1 = 0 \). To solve such an equation:

• Apply algebraic methods such as factoring or the Quadratic Formula.
• Find the trigonometric values (like \( \tan x \)) that solve the equation.
• Determine the corresponding angle(s) \( x \) using inverse trigonometric functions (often \( \arctan(x) \)).
• Limit the solutions to a specified interval, here \([0, 2\pi)\).
• Use graphing utilities to visually verify solutions, offering a practical check against arithmetic errors.

By breaking down the equation into manageable parts, you develop a better grasp of the solution process. Understanding these steps ensures confidence when approaching similar trigonometric equations, helping to uncover the desired solutions efficiently.