Problem 84 - Precalculus Essentials

Question

Use an identity to solve each equation on the interval \([0,2 \pi)\) $$\tan x-\sec x=1$$

Step-by-Step Solution

Answer

The short answer depends on the solutions of the quadratic equation. Following the steps outlined above will yield the correct solutions.

Step 1: Rewrite as familiar trigonometric functions

To solve this equation, first we have to rewrite \(\tan x\) and \(\sec x\) in terms of more familiar trigonometric functions. We know that \(\tan x = \frac{\sin x}{\cos x}\) and \(\sec x = \frac{1}{\cos x}\). Hence, the equation will take the form \(\frac{\sin x}{\cos x} - \frac{1}{\cos x} = 1\)

Step 2: Simplify the Equation

After rewriting, we can simplify the equation by finding the common denominator which is \(\cos x\), then subtracting \(\sin x\) from 1. This results in the equation \(\frac{1-\sin x}{\cos x} = 1\)

Step 3: Cross-multiply and Simplify

To further simplify the equation, we can cross-multiply to get rid of the fraction. This results into \(\cos x = 1 - \sin x\)

Step 4: Square both sides of the Equation

To get a quadratic equation, we can square both sides of the equation which results in \(\cos^2 x = (1 - \sin x)^2\). We know that \(\cos^2 x = 1 - \sin^2 x\), hence \(1 - \sin^2 x = 1 - 2\sin x + \sin^2 x\)

Step 5: Simplify and solve the equation

Solving the equation further, we get \(-2 \sin x + 1 = 2 \sin^2 x \), or \(2 \sin^2 x + 2 \sin x - 1 = 0\). This is a quadratic equation in terms of \(\sin x\). You can then solve this quadratic equation to find the possible values of \(\sin x\) and substitute these values into the original equation to find the corresponding values of \(x\). Remember, the solution should fall within the domain \([0, 2\pi)\)

Step 6: Apply the sin function and interval restrictions

Finally, apply the inverse \(\sin\) function to find the possible values of \(x\) and ensure the solutions lie in the interval \([0, 2\pi)\).

Key Concepts

Solving Trigonometric Equations

When dealing with trigonometric equations like \( \tan x - \sec x = 1 \), the first step is usually to express all the trigonometric functions in terms of the basic sine and cosine functions. This often simplifies the problem, making it easier to apply various identities or algebraic methods. Here, by replacing \( \tan x = \frac{\sin x}{\cos x} \) and \( \sec x = \frac{1}{\cos x} \), we reformulate the equation to a more manageable form. Once identities have been applied, the next step is to solve the equation, often by simplifying it further, or by isolating one of the variables.

• Express trigonometric functions in terms of sine and cosine.
• Simplify the equation using algebraic manipulations like finding a common denominator.

Finally, solve for the unknown variable while keeping in mind the specific interval or domain of the angle provided in the question.

Interval Notation

Interval notation is a method of defining a set of numbers along a continuous range. In trigonometric equations, we often see problems that require finding solutions within a specified interval, such as \([0, 2\pi)\). This means we are looking for all possible values of \(x\) within one complete cycle of a circle, but not including \(2\pi\) itself.

• The interval \([0, 2\pi)\) indicates that solutions include 0 but not \(2\pi\).
• Consideration of the interval is crucial because trigonometric functions are periodic, meaning they repeat their values over intervals.

Therefore, once the equation is solved, any solutions must be verified within the given interval to ensure accuracy.

Quadratic Equations in Trigonometry

Sometimes, solving a trigonometric equation involves transforming it into a quadratic equation. This occurs when squaring both sides of a trigonometric equation. For instance, if we reach an equation like \( \cos^2 x = (1 - \sin x)^2 \), knowing that \( \cos^2 x = 1 - \sin^2 x \) helps transform it into a quadratic equation in \( \sin x \).

• Utilize trigonometric identities to manipulate the form of the equation.
• Express the equation fully as a standard quadratic form, such as \( ax^2 + bx + c = 0 \).

Solving this will yield one or more values for \( \sin x \), which can then be used to find solutions to the original trigonometric equation.

Trigonometric Functions Transformation

Transformations of trigonometric functions are often involved in solving trigonometric equations. By changing the form of the function using identities, you can make the calculus much simpler. This often includes using identities like \( \sec x = \frac{1}{\cos x} \) or recognizing that \( \tan x = \frac{\sin x}{\cos x} \) to simplify equations directly.

• Applying known identities can convert complex functions into more familiar forms.
• These transformations lead to more straightforward algebraic manipulation.

By doing so, not only does it simplify solving the equation, but it also helps in visualizing the transformations on a graph for a deeper understanding of the problem.