Solving 3 Sec X - 1 = 2 On [0, 2π] A Comprehensive Guide

Trigonometric equations are fundamental in various fields of mathematics, physics, and engineering. These equations involve trigonometric functions such as sine, cosine, tangent, secant, cosecant, and cotangent. Solving these equations often requires a blend of algebraic manipulation, trigonometric identities, and a solid understanding of the unit circle. In this article, we delve into a step-by-step approach to solve the trigonometric equation $3 ext{sec } x - 1 = 2$ within the interval $[0, 2\[Pi]]$. This interval represents one full revolution around the unit circle, ensuring we capture all possible solutions within a single period.

Understanding Trigonometric Equations

Before diving into the specifics, it's crucial to understand what makes trigonometric equations unique. Unlike algebraic equations that deal with polynomial expressions, trigonometric equations involve functions that oscillate and repeat their values at regular intervals. This periodicity means that a trigonometric equation can have infinitely many solutions. However, when we restrict the domain to a specific interval, such as $[0, 2\[Pi]]$, we limit the solutions to a finite set.

Key Trigonometric Functions

The primary trigonometric functions are sine ($\sin x$), cosine ($\cos x$), and tangent ($\tan x$). Their reciprocals are cosecant ($\csc x = 1/\sin x$), secant ($\sec x = 1/\cos x$), and cotangent ($\cot x = 1/\tan x$), respectively. These functions relate angles in a right triangle to the ratios of its sides. On the unit circle, where the radius is 1, the coordinates of a point on the circle corresponding to an angle $x$ are given by $(\cos x, \sin x)$. This geometric interpretation is invaluable for solving trigonometric equations.

The Unit Circle

The unit circle is an essential tool for visualizing and understanding trigonometric functions. It's a circle with a radius of 1 centered at the origin in the Cartesian plane. Angles are measured counterclockwise from the positive x-axis. The x-coordinate of a point on the unit circle represents the cosine of the angle, and the y-coordinate represents the sine of the angle. The unit circle helps us identify angles that have specific trigonometric values, which is crucial for solving equations.

General Strategies for Solving Trigonometric Equations

Solving trigonometric equations often involves the following steps:

1. Isolate the Trigonometric Function: Use algebraic manipulations to isolate the trigonometric function on one side of the equation.
2. Identify Angles: Determine the angles within the given interval that satisfy the isolated trigonometric function.
3. Use the Unit Circle and Trigonometric Identities: Utilize the unit circle and trigonometric identities to find all possible solutions.
4. Check for Extraneous Solutions: Verify that the solutions obtained are valid within the given domain and do not lead to undefined expressions.

Solving the Equation $3 \sec x - 1 = 2$

Now, let's apply these strategies to solve the equation $3 \sec x - 1 = 2$ on the interval $[0, 2\[Pi]]$.

Step 1: Isolate the Trigonometric Function

Our first goal is to isolate the secant function. We begin by adding 1 to both sides of the equation:

$3 \sec x - 1 + 1 = 2 + 1$

$3 \sec x = 3$

Next, we divide both sides by 3:

$\frac{3 \sec x}{3} = \frac{3}{3}$

$\sec x = 1$

Now we have isolated the secant function. This simplified equation is much easier to work with.

Step 2: Rewrite in Terms of Cosine

Since $\sec x$ is the reciprocal of $\cos x$, we can rewrite the equation in terms of cosine:

$\sec x = \frac{1}{\cos x}$

So, our equation $\sec x = 1$ becomes:

$\frac{1}{\cos x} = 1$

To solve for $\cos x$, we can take the reciprocal of both sides:

$\cos x = 1$

Step 3: Identify Angles on the Unit Circle

Now we need to find the angles $x$ in the interval $[0, 2\[Pi]]$ where $\cos x = 1$. Recall that on the unit circle, the cosine of an angle is represented by the x-coordinate of the point on the circle.

We are looking for points on the unit circle where the x-coordinate is 1. This occurs at the angle $x = 0$ radians (or $0$ degrees).

As we continue around the unit circle, we complete one full revolution at $x = 2\[Pi]$ radians (or $360$ degrees). At this point, the cosine is also 1. However, since our interval is $[0, 2\[Pi])$, we include $0$ but exclude $2\[Pi]$ if the interval was $[0, 2\[Pi])$. In this case, $2\[Pi]$ can also be a solution since the interval is inclusive. However, the angle $2\[Pi]$ is coterminal with the angle $0$, so it represents the same point on the unit circle.

Therefore, within the interval $[0, 2\[Pi]]$, there is only one unique solution: $x = 0$.

Step 4: Verify the Solution

To ensure our solution is correct, we substitute $x = 0$ back into the original equation:

$3 \sec(0) - 1 = 2$

Since $\sec(0) = \frac{1}{\cos(0)}$ and $\cos(0) = 1$, we have:

$3 \cdot \frac{1}{1} - 1 = 2$

$3 - 1 = 2$

$2 = 2$

The equation holds true, so our solution $x = 0$ is valid.

Conclusion

In summary, the solution to the trigonometric equation $3 \sec x - 1 = 2$ on the interval $[0, 2\[Pi]]$ is $x = 0$. This process involved isolating the trigonometric function, converting it to cosine, identifying angles on the unit circle, and verifying the solution. Understanding the unit circle and trigonometric identities is crucial for solving these types of equations.

Solving trigonometric equations requires a systematic approach. By following these steps, we can confidently find solutions within specified intervals. The unit circle serves as a visual aid, allowing us to connect angles with their corresponding trigonometric values. This method not only solves the equation but also reinforces a deeper understanding of trigonometric functions and their properties. Therefore, the correct answer is:

B. 0

Additional Practice

To reinforce your understanding, try solving similar trigonometric equations. For example:

1. $2 \sin x + 1 = 0$ on $[0, 2\[Pi]]$
2. $\tan x - 1 = 0$ on $[0, 2\[Pi]]$
3. $4 \cos x - 2 = 0$ on $[0, 2\[Pi]]$

By practicing various problems, you will become more proficient in solving trigonometric equations and gain a deeper appreciation for the beauty and utility of trigonometry.

This comprehensive guide provides a detailed explanation of how to solve the given trigonometric equation, emphasizing the importance of understanding the unit circle and trigonometric identities. Through a step-by-step approach, we've not only found the solution but also reinforced the fundamental principles of trigonometry.