Solving Trigonometric Equations: 2sin3x - 1 0

Introduction

Trigonometric equations are a fundamental part of mathematics, appearing in various applications from physics to engineering. This article will guide you through solving a specific trigonometric equation: 2sin3x - 1 0. We will break down the process step-by-step, providing a clear understanding of each stage.

Step-by-Step Solution

Step 1: Isolate the Sine Function

Our starting equation is: 2sin3x - 1 0

First, let’s isolate the sine function by adding 1 to both sides of the equation:

2sin3x 1

Then, divide both sides by 2:

sin3x 1/2

Step 2: Find General Solutions for sin3x 1/2

Next, we need to find the general solutions for the equation sin3x 1/2. Since the sine function equals 1/2 at certain angles, we have:

3x π/6 2kπ and 3x 5π/6 2kπ

Where k is any integer. This is because the sine function is periodic with a period of 2π.

Step 3: Solve for x

To determine x, we divide both sides of the equations by 3:

x π/18 2kπ/3 and x 5π/18 2kπ/3

Therefore, the complete solution set is:

x π/18 2kπ/3 and x 5π/18 2kπ/3

Where k is any integer. This gives us all the values of x that satisfy the original equation.

Simplified Solution for Different Reference Angles

For a more streamlined approach, we can simplify our solution set by considering different reference angles for the sine function. Specifically, if sin3x 1/2, then:

3x u03C0/6 2ku03C0 and 3x 5u03C0/6 2ku03C0

Let's break it down further:

x u03C0/18 2ku03C0/3 and x 5u03C0/18 2ku03C0/3

Where k is any integer.

Alternative Methods and Considerations

Another approach, often seen in simpler contexts, involves using known values of sine. Since the sine of 30 degrees (or u03C0/6 radians) is 1/2, we can directly infer:

3x u03C0/6 or 3x 5u03C0/6

Solving these gives us:

x u03C0/18 or x 5u03C0/18

These solutions are valid for any integer k in the general set ku03C0/3 u00B1 u03C0/18.

Conclusion

Understanding the steps to solve trigonometric equations such as 2sin3x - 1 0 is crucial for advanced mathematical applications. By breaking down the problem into manageable steps and considering various approaches, we can ensure a thorough understanding and correct solutions.