Determine The General Solution Of The Equation:$ 6 \cos X - 5 = \frac{4}{\cos X} }$Note { \cos X \neq 0$ $.

Mar 11, 2025 by ADMIN 109 views

**Solving Trigonometric Equations: A Step-by-Step Guide to Determining the General Solution**

Introduction

Trigonometric equations are a fundamental part of mathematics, and solving them requires a deep understanding of trigonometric functions and their properties. In this article, we will focus on solving a specific type of trigonometric equation, namely the equation $6 \cos x - 5 = \frac{4}{\cos x}$ . This equation involves the cosine function and is subject to the condition $\cos x \neq 0$ . Our goal is to determine the general solution of this equation, which will involve various trigonometric identities and techniques.

Understanding the Equation

The given equation is $6 \cos x - 5 = \frac{4}{\cos x}$ . To begin solving this equation, we need to understand its structure and the properties of the trigonometric functions involved. The equation involves the cosine function, which is a periodic function that oscillates between -1 and 1. The equation also involves the reciprocal of the cosine function, which is $\frac{1}{\cos x}$ . This reciprocal function is also periodic and oscillates between -1 and 1.

Simplifying the Equation

To simplify the equation, we can start by multiplying both sides of the equation by $\cos x$ . This will eliminate the fraction on the right-hand side of the equation. Multiplying both sides by $\cos x$ gives us:

6 \cos^2 x - 5 \cos x = 4

Using Trigonometric Identities

The next step is to use trigonometric identities to simplify the equation further. We can use the Pythagorean identity, which states that $\sin^2 x + \cos^2 x = 1$ . Since we have $\cos^2 x$ in the equation, we can use this identity to rewrite the equation in terms of $\sin^2 x$ .

Rewriting the Equation

Using the Pythagorean identity, we can rewrite the equation as:

6 (1 - \sin^2 x) - 5 \cos x = 4

Simplifying the Equation

Simplifying the equation further, we get:

6 - 6 \sin^2 x - 5 \cos x = 4

Rearranging the Equation

Rearranging the equation, we get:

-6 \sin^2 x - 5 \cos x = -2

Using Trigonometric Identities Again

We can use the Pythagorean identity again to rewrite the equation in terms of $\sin x$ and $\cos x$ . This gives us:

-6 \sin^2 x - 5 \cos x = -2

Solving for $\sin x$

To solve for $\sin x$ , we can use the quadratic formula. Rearranging the equation, we get:

6 \sin^2 x + 5 \cos x + 2 = 0

Using the Quadratic Formula

Using the quadratic formula, we get:

\sin x = \frac{-5 \cos x - 2}{6 \sin x}

Simplifying the Equation

Simplifying the equation further, we get:

6 \sin^2 x + 5 \cos x + 2 = 0

Factoring the Equation

Factoring the equation, we get:

(2 \sin x + 1)(3 \sin x + 2) = 0

Solving for $\sin x$

Solving for $\sin x$ , we get:

\sin x = -\frac{1}{2} \text{ or } \sin x = -\frac{2}{3}

Finding the General Solution

To find the general solution, we need to find the values of $x$ that satisfy the equation. We can use the inverse sine function to find the values of $x$ that satisfy the equation.

Using the Inverse Sine Function

Using the inverse sine function, we get:

x = \sin^{-1} \left( -\frac{1}{2} \right) \text{ or } x = \sin^{-1} \left( -\frac{2}{3} \right)

Finding the General Solution

The general solution is:

x = \sin^{-1} \left( -\frac{1}{2} \right) + 2 \pi n \text{ or } x = \sin^{-1} \left( -\frac{2}{3} \right) + 2 \pi n

where $n$ is an integer.

Conclusion

In this article, we have solved a trigonometric equation involving the cosine function and its reciprocal. We have used various trigonometric identities and techniques to simplify the equation and find the general solution. The general solution is given by:

x = \sin^{-1} \left( -\frac{1}{2} \right) + 2 \pi n \text{ or } x = \sin^{-1} \left( -\frac{2}{3} \right) + 2 \pi n

where $n$ is an integer.

Final Answer

The final answer is:

x = \sin^{-1} \left( -\frac{1}{2} \right) + 2 \pi n \text{ or } x = \sin^{-1} \left( -\frac{2}{3} \right) + 2 \pi n$<br/> # **Solving Trigonometric Equations: A Q&A Guide**

Introduction

In our previous article, we solved a trigonometric equation involving the cosine function and its reciprocal. We used various trigonometric identities and techniques to simplify the equation and find the general solution. In this article, we will provide a Q&A guide to help you understand the solution and apply it to similar problems.

Q: What is the general solution of the equation $6 \cos x - 5 = \frac{4}{\cos x}$ ?

A: The general solution is:

Introduction

Understanding the Equation

Simplifying the Equation

Using Trigonometric Identities

Rewriting the Equation

Simplifying the Equation

Rearranging the Equation

Using Trigonometric Identities Again

Solving for sin⁡x\sin xsinx

Using the Quadratic Formula

Simplifying the Equation

Factoring the Equation

Solving for sin⁡x\sin xsinx

Finding the General Solution

Using the Inverse Sine Function

Finding the General Solution

Conclusion

Final Answer

Introduction

Q: What is the general solution of the equation 6cos⁡x−5=4cos⁡x6 \cos x - 5 = \frac{4}{\cos x}6cosx−5=cosx4​?

Solving for $\sin x$

Solving for $\sin x$

Q: What is the general solution of the equation $6 \cos x - 5 = \frac{4}{\cos x}$ ?