Find The General Solution Of $5 \sin X - 2 = 1 - \cos 2x$.

Mar 12, 2025 by ADMIN 61 views

**Solving Trigonometric Equations: A Step-by-Step Guide**

Introduction

Trigonometric equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on finding the general solution of a specific trigonometric equation, which involves sine and cosine functions. We will break down the solution into manageable steps, making it easier to understand and apply.

The Given Equation

The given equation is:

5 \sin x - 2 = 1 - \cos 2x

Our goal is to find the general solution of this equation, which means we need to isolate the trigonometric functions and express the equation in terms of a single variable.

Step 1: Simplify the Equation

To simplify the equation, we can start by isolating the sine and cosine terms. We can rewrite the equation as:

5 \sin x = 1 + \cos 2x + 2

Simplifying further, we get:

5 \sin x = 3 + \cos 2x

Step 2: Use Trigonometric Identities

To make the equation more manageable, we can use trigonometric identities to simplify the cosine term. We know that:

\cos 2x = 1 - 2 \sin^2 x

Substituting this identity into the equation, we get:

5 \sin x = 3 + 1 - 2 \sin^2 x

Simplifying further, we get:

5 \sin x = 4 - 2 \sin^2 x

Step 3: Rearrange the Equation

To isolate the sine term, we can rearrange the equation as follows:

2 \sin^2 x + 5 \sin x - 4 = 0

This is a quadratic equation in terms of the sine function.

Step 4: Solve the Quadratic Equation

To solve the quadratic equation, we can use the quadratic formula:

\sin x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

In this case, $a = 2$ , $b = 5$ , and $c = -4$ . Plugging these values into the formula, we get:

\sin x = \frac{-5 \pm \sqrt{5^2 - 4(2)(-4)}}{2(2)}

Simplifying further, we get:

\sin x = \frac{-5 \pm \sqrt{25 + 32}}{4}

\sin x = \frac{-5 \pm \sqrt{57}}{4}

Step 5: Find the General Solution

To find the general solution, we need to consider both the positive and negative square roots. We can write the general solution as:

\sin x = \frac{-5 + \sqrt{57}}{4} \quad \text{or} \quad \sin x = \frac{-5 - \sqrt{57}}{4}

Using the inverse sine function, we can find the general solution:

x = \sin^{-1} \left( \frac{-5 + \sqrt{57}}{4} \right) + 2 \pi n \quad \text{or} \quad x = \sin^{-1} \left( \frac{-5 - \sqrt{57}}{4} \right) + 2 \pi n

where $n$ is an integer.

Conclusion

In this article, we have solved a trigonometric equation involving sine and cosine functions. We have broken down the solution into manageable steps, making it easier to understand and apply. The general solution of the equation is given by:

x = \sin^{-1} \left( \frac{-5 + \sqrt{57}}{4} \right) + 2 \pi n \quad \text{or} \quad x = \sin^{-1} \left( \frac{-5 - \sqrt{57}}{4} \right) + 2 \pi n

where $n$ is an integer.

Final Answer

The final answer is:

x = \sin^{-1} \left( \frac{-5 + \sqrt{57}}{4} \right) + 2 \pi n \quad \text{or} \quad x = \sin^{-1} \left( \frac{-5 - \sqrt{57}}{4} \right) + 2 \pi n$<br/> **Solving Trigonometric Equations: A Q&A Guide** =====================================================

Introduction

In our previous article, we solved a trigonometric equation involving sine and cosine functions. We broke down the solution into manageable steps, making it easier to understand and apply. In this article, we will provide a Q&A guide to help you better understand the solution and apply it to similar problems.

Q: What is the general solution of the equation?

A: The general solution of the equation is given by: