Solve For $x$ Where $0 \leq X \leq \frac \pi}{2}$ $4 \sqrt{3 \csc X - 4 - 3 \csc^2 X = 0$

Introduction

Trigonometric equations are a fundamental aspect of mathematics, and solving them requires a deep understanding of trigonometric functions and identities. In this article, we will focus on solving a specific trigonometric equation involving the cosecant function. The given equation is:

$$4 \sqrt{3} \csc x - 4 - 3 \csc^2 x = 0$$

where $0 \leq x \leq \frac{\pi}{2}$. Our goal is to find the value of $x$ that satisfies this equation.

Understanding the Cosecant Function

Before we dive into solving the equation, let's take a moment to understand the cosecant function. The cosecant function, denoted by $\csc x$, is the reciprocal of the sine function. In other words:

$$\csc x = \frac{1}{\sin x}$$

The cosecant function has a period of $2\pi$ and is positive in the first and fourth quadrants.

Solving the Equation

To solve the given equation, we can start by rearranging the terms to isolate the cosecant function:

$$4 \sqrt{3} \csc x - 3 \csc^2 x = 4$$

Next, we can factor out the common term $\csc x$:

$$\csc x (4 \sqrt{3} - 3 \csc x) = 4$$

Now, we can divide both sides by $\csc x$ to get:

$$4 \sqrt{3} - 3 \csc x = \frac{4}{\csc x}$$

Simplifying further, we get:

$$4 \sqrt{3} - 3 \csc x = 4 \sin x$$

Now, we can rearrange the terms to isolate the cosecant function:

$$3 \csc x = 4 \sin x - 4 \sqrt{3}$$

Dividing both sides by $3$, we get:

$$\csc x = \frac{4 \sin x - 4 \sqrt{3}}{3}$$

Using the Pythagorean Identity

We can use the Pythagorean identity to rewrite the equation in terms of the sine function:

$$\sin^2 x + \cos^2 x = 1$$

Rearranging the terms, we get:

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

Substituting this expression into the previous equation, we get:

$$\csc x = \frac{4 \sqrt{1 - \cos^2 x} - 4 \sqrt{3}}{3}$$

Simplifying the Expression

We can simplify the expression by using the fact that $\sin^2 x = 1 - \cos^2 x$:

$$\csc x = \frac{4 \sqrt{1 - \cos^2 x} - 4 \sqrt{3}}{3}$$

$$\csc x = \frac{4 \sqrt{\sin^2 x} - 4 \sqrt{3}}{3}$$

$$\csc x = \frac{4 \sin x - 4 \sqrt{3}}{3}$$

Finding the Value of x

Now that we have simplified the expression, we can find the value of $x$ that satisfies the equation. We can start by setting the expression equal to $\csc x$:

$$\frac{4 \sin x - 4 \sqrt{3}}{3} = \csc x$$

Multiplying both sides by $3$, we get:

$$4 \sin x - 4 \sqrt{3} = 3 \csc x$$

Now, we can substitute the expression for $\csc x$:

$$4 \sin x - 4 \sqrt{3} = 3 \left(\frac{4 \sin x - 4 \sqrt{3}}{3}\right)$$

Simplifying further, we get:

$$4 \sin x - 4 \sqrt{3} = 4 \sin x - 4 \sqrt{3}$$

This equation is true for all values of $x$, so we can conclude that:

$$x = \sin^{-1} \left(\frac{4 \sqrt{3}}{4}\right)$$

$$x = \sin^{-1} \left(\sqrt{3}\right)$$

Conclusion

In this article, we solved a trigonometric equation involving the cosecant function. We started by rearranging the terms to isolate the cosecant function, and then used the Pythagorean identity to rewrite the equation in terms of the sine function. We simplified the expression and found the value of $x$ that satisfies the equation. The final answer is:

$$x = \sin^{-1} \left(\sqrt{3}\right)$$

Final Answer

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Introduction

In our previous article, we solved a trigonometric equation involving the cosecant function. In this article, we will provide a Q&A guide to help you better understand the concepts and techniques involved in solving trigonometric equations.

Q: What is a trigonometric equation?

A: A trigonometric equation is an equation that involves trigonometric functions, such as sine, cosine, and tangent. These equations can be used to model real-world problems, such as the motion of objects, the behavior of electrical circuits, and the properties of waves.

Q: What are the common trigonometric functions?

A: The common trigonometric functions are:

• Sine (sin x)
• Cosine (cos x)
• Tangent (tan x)
• Cosecant (csc x)
• Secant (sec x)
• Cotangent (cot x)

Q: How do I solve a trigonometric equation?

A: To solve a trigonometric equation, you can follow these steps:

1. Simplify the equation by combining like terms and using trigonometric identities.
2. Isolate the trigonometric function by moving all other terms to one side of the equation.
3. Use the inverse trigonometric function to find the value of the variable.
4. Check your solution by plugging it back into the original equation.

Q: What are some common trigonometric identities?

A: Some common trigonometric identities include:

• sin^2 x + cos^2 x = 1
• tan x = sin x / cos x
• cot x = cos x / sin x
• sec x = 1 / cos x
• csc x = 1 / sin x

Q: How do I use the Pythagorean identity?

A: The Pythagorean identity is a fundamental identity in trigonometry that states:

sin^2 x + cos^2 x = 1

You can use this identity to rewrite the equation in terms of the sine or cosine function.

Q: What is the inverse trigonometric function?

A: The inverse trigonometric function is a function that takes a value of a trigonometric function and returns the angle that produces that value. For example, the inverse sine function takes a value of the sine function and returns the angle that produces that value.

Q: How do I use the inverse trigonometric function?

A: To use the inverse trigonometric function, you can follow these steps:

1. Identify the trigonometric function that is involved in the equation.
2. Use the inverse trigonometric function to find the value of the variable.
3. Check your solution by plugging it back into the original equation.

Q: What are some common mistakes to avoid when solving trigonometric equations?

A: Some common mistakes to avoid when solving trigonometric equations include:

• Not simplifying the equation enough
• Not isolating the trigonometric function
• Not using the correct inverse trigonometric function
• Not checking the solution

Conclusion

In this article, we provided a Q&A guide to help you better understand the concepts and techniques involved in solving trigonometric equations. We covered topics such as trigonometric functions, identities, and inverse trigonometric functions. We also provided tips and tricks for solving trigonometric equations and avoiding common mistakes.

Final Answer

The final answer is $\boxed{\frac{\pi}{3}}$.