Find All Solutions Of The Equation In The Interval \[0, 2\pi$\].$\csc X (2 \cos X + 1) = 0$Write Your Answer In Radians In Terms Of $\pi$. If There Is More Than One Solution, Separate Them With Commas.$x =$

Introduction

In this article, we will delve into the world of trigonometric equations and explore the solutions to the given equation $\csc x (2 \cos x + 1) = 0$ in the interval $[0, 2\pi]$. The equation involves the cosecant and cosine functions, making it a challenging problem that requires a thorough understanding of trigonometric identities and properties.

Understanding the Equation

The given equation is $\csc x (2 \cos x + 1) = 0$. To solve this equation, we need to find the values of $x$ that satisfy the equation. We can start by analyzing the properties of the cosecant and cosine functions.

The cosecant function is defined as $\csc x = \frac{1}{\sin x}$, and the cosine function is defined as $\cos x = \frac{e^{ix} + e^{-ix}}{2}$. Using these definitions, we can rewrite the equation as $\frac{1}{\sin x} (2 \cos x + 1) = 0$.

Solving for $\sin x$

To solve the equation, we need to find the values of $x$ that make the expression $\frac{1}{\sin x} (2 \cos x + 1)$ equal to zero. This means that either $\sin x = 0$ or $2 \cos x + 1 = 0$.

Solving for $\sin x = 0$

The sine function is equal to zero at $x = 0, \pi, 2\pi$. These are the values of $x$ that satisfy the equation $\sin x = 0$.

Solving for $2 \cos x + 1 = 0$

To solve the equation $2 \cos x + 1 = 0$, we can isolate the cosine function by subtracting 1 from both sides and then dividing by 2. This gives us $\cos x = -\frac{1}{2}$.

The cosine function is equal to $-\frac{1}{2}$ at $x = \frac{2\pi}{3}, \frac{4\pi}{3}$. These are the values of $x$ that satisfy the equation $2 \cos x + 1 = 0$.

Combining the Solutions

We have found the values of $x$ that satisfy the equation $\sin x = 0$ and the equation $2 \cos x + 1 = 0$. To find the final solutions, we need to combine these values.

The values of $x$ that satisfy the equation $\sin x = 0$ are $x = 0, \pi, 2\pi$. The values of $x$ that satisfy the equation $2 \cos x + 1 = 0$ are $x = \frac{2\pi}{3}, \frac{4\pi}{3}$.

Conclusion

In this article, we have solved the equation $\csc x (2 \cos x + 1) = 0$ in the interval $[0, 2\pi]$. We have found the values of $x$ that satisfy the equation by analyzing the properties of the cosecant and cosine functions. The final solutions are $x = 0, \frac{2\pi}{3}, \pi, \frac{4\pi}{3}, 2\pi$.

Final Answer

Q&A: Frequently Asked Questions

Q: What is the equation $\csc x (2 \cos x + 1) = 0$? A: The equation $\csc x (2 \cos x + 1) = 0$ is a trigonometric equation that involves the cosecant and cosine functions. It is a challenging problem that requires a thorough understanding of trigonometric identities and properties.

Q: How do I solve the equation $\csc x (2 \cos x + 1) = 0$? A: To solve the equation, we need to find the values of $x$ that satisfy the equation. We can start by analyzing the properties of the cosecant and cosine functions. We can rewrite the equation as $\frac{1}{\sin x} (2 \cos x + 1) = 0$ and then solve for $\sin x = 0$ or $2 \cos x + 1 = 0$.

Q: What are the values of $x$ that satisfy the equation $\sin x = 0$? A: The values of $x$ that satisfy the equation $\sin x = 0$ are $x = 0, \pi, 2\pi$. These are the values of $x$ that make the sine function equal to zero.

Q: What are the values of $x$ that satisfy the equation $2 \cos x + 1 = 0$? A: The values of $x$ that satisfy the equation $2 \cos x + 1 = 0$ are $x = \frac{2\pi}{3}, \frac{4\pi}{3}$. These are the values of $x$ that make the cosine function equal to $-\frac{1}{2}$.

Q: How do I combine the solutions to find the final answer? A: To find the final answer, we need to combine the values of $x$ that satisfy the equation $\sin x = 0$ and the equation $2 \cos x + 1 = 0$. The final solutions are $x = 0, \frac{2\pi}{3}, \pi, \frac{4\pi}{3}, 2\pi$.

Q: What is the final answer to the equation $\csc x (2 \cos x + 1) = 0$? A: The final answer to the equation $\csc x (2 \cos x + 1) = 0$ is $x = 0, \frac{2\pi}{3}, \pi, \frac{4\pi}{3}, 2\pi$.

Q: Why is it important to understand trigonometric equations? A: Understanding trigonometric equations is important because it helps us solve problems in mathematics, physics, engineering, and other fields. Trigonometric equations are used to model real-world problems, such as the motion of objects, the behavior of waves, and the properties of periodic functions.

Q: How can I practice solving trigonometric equations? A: You can practice solving trigonometric equations by working through examples and exercises in your textbook or online resources. You can also try solving problems on your own and then checking your answers with a calculator or online tool.

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

• Not using the correct trigonometric identities or formulas
• Not simplifying the equation correctly
• Not checking the solutions to make sure they are valid
• Not using a calculator or online tool to check the solutions

Q: How can I improve my skills in solving trigonometric equations? A: You can improve your skills in solving trigonometric equations by:

• Practicing regularly and working through examples and exercises
• Using a calculator or online tool to check your solutions
• Asking for help from a teacher or tutor if you are struggling
• Reviewing the basics of trigonometry, including the definitions of the trigonometric functions and the properties of the unit circle.