Find All Solutions Of The Equation 2 Cos ⁡ 3 X = 1 2 \cos 3x = 1 2 Cos 3 X = 1 In The Interval {0, \pi }$.The Answers Are X 1 = □ X_1 = \square X 1 ​ = □ , X 2 = □ X_2 = \square X 2 ​ = □ , And X 3 = □ X_3 = \square X 3 ​ = □ With X 1 \textless X 2 \textless X 3 X_1 \ \textless \ X_2 \ \textless \ X_3 X 1 ​ \textless X 2 ​ \textless X 3 ​ .

Introduction

In this article, we will delve into the world of trigonometry and solve the equation $2 \cos 3x = 1$ in the interval $[0, \pi]$. This equation involves the cosine function, which is a fundamental concept in mathematics. We will use various techniques to find the solutions of this equation and provide a step-by-step guide to help readers understand the process.

Understanding the Equation

The given equation is $2 \cos 3x = 1$. To solve this equation, we need to isolate the cosine function. We can start by dividing both sides of the equation by 2, which gives us $\cos 3x = \frac{1}{2}$.

Recalling the Cosine Function

The cosine function is a periodic function that oscillates between -1 and 1. The value of $\cos \theta$ is equal to 1 when $\theta$ is a multiple of $2\pi$, and it is equal to -1 when $\theta$ is a multiple of $\pi + 2\pi$. In this case, we are interested in finding the values of $x$ for which $\cos 3x = \frac{1}{2}$.

Finding the Solutions

To find the solutions of the equation $\cos 3x = \frac{1}{2}$, we need to recall the unit circle and the values of the cosine function at different angles. We know that $\cos \theta = \frac{1}{2}$ when $\theta$ is equal to $\frac{\pi}{3}$ or $\frac{5\pi}{3}$. Since we have $3x$ in the equation, we need to divide these angles by 3 to get the values of $x$.

Solving for $x$

We can now solve for $x$ by dividing the angles by 3. We get $x = \frac{\pi}{9}$ and $x = \frac{5\pi}{9}$. However, we need to find the solutions in the interval $[0, \pi]$. We can do this by finding the values of $x$ that satisfy the equation in this interval.

Finding the Solutions in the Interval $[0, \pi]$

We can start by finding the values of $x$ that satisfy the equation in the interval $[0, \frac{\pi}{2}]$. We know that $\cos 3x = \frac{1}{2}$ when $3x = \frac{\pi}{3}$ or $3x = \frac{5\pi}{3}$. We can divide these angles by 3 to get the values of $x$. We get $x = \frac{\pi}{9}$ and $x = \frac{5\pi}{9}$.

Finding the Second Solution

We need to find the second solution in the interval $[0, \pi]$. We can do this by finding the value of $x$ that satisfies the equation in the interval $[\frac{\pi}{2}, \pi]$. We know that $\cos 3x = \frac{1}{2}$ when $3x = \frac{7\pi}{3}$ or $3x = \frac{11\pi}{3}$. We can divide these angles by 3 to get the values of $x$. We get $x = \frac{7\pi}{9}$ and $x = \frac{11\pi}{9}$.

Finding the Third Solution

We need to find the third solution in the interval $[0, \pi]$. We can do this by finding the value of $x$ that satisfies the equation in the interval $[\frac{\pi}{2}, \pi]$. We know that $\cos 3x = \frac{1}{2}$ when $3x = \frac{13\pi}{3}$ or $3x = \frac{17\pi}{3}$. We can divide these angles by 3 to get the values of $x$. We get $x = \frac{13\pi}{9}$ and $x = \frac{17\pi}{9}$.

Conclusion

In this article, we have solved the equation $2 \cos 3x = 1$ in the interval $[0, \pi]$. We have found three solutions: $x_1 = \frac{\pi}{9}$, $x_2 = \frac{7\pi}{9}$, and $x_3 = \frac{13\pi}{9}$. These solutions satisfy the equation in the interval $[0, \pi]$.

Final Answer

The final answer is:

$x_1 = \frac{\pi}{9}$ $x_2 = \frac{7\pi}{9}$ $x_3 = \frac{13\pi}{9}$

Q: What is the equation $2 \cos 3x = 1$?

A: The equation $2 \cos 3x = 1$ is a trigonometric equation that involves the cosine function. It is a quadratic equation in terms of the cosine function, and we need to solve for the values of $x$ that satisfy this equation in the interval $[0, \pi]$.

Q: How do we solve the equation $2 \cos 3x = 1$?

A: To solve the equation $2 \cos 3x = 1$, we need to isolate the cosine function. We can start by dividing both sides of the equation by 2, which gives us $\cos 3x = \frac{1}{2}$. We can then use the unit circle and the values of the cosine function at different angles to find the solutions.

Q: What are the solutions to the equation $2 \cos 3x = 1$ in the interval $[0, \pi]$?

A: The solutions to the equation $2 \cos 3x = 1$ in the interval $[0, \pi]$ are $x_1 = \frac{\pi}{9}$, $x_2 = \frac{7\pi}{9}$, and $x_3 = \frac{13\pi}{9}$.

Q: How do we find the second solution in the interval $[0, \pi]$?

A: To find the second solution in the interval $[0, \pi]$, we need to find the value of $x$ that satisfies the equation in the interval $[\frac{\pi}{2}, \pi]$. We can do this by finding the value of $x$ that satisfies the equation when $3x = \frac{7\pi}{3}$ or $3x = \frac{11\pi}{3}$.

Q: How do we find the third solution in the interval $[0, \pi]$?

A: To find the third solution in the interval $[0, \pi]$, we need to find the value of $x$ that satisfies the equation in the interval $[\frac{\pi}{2}, \pi]$. We can do this by finding the value of $x$ that satisfies the equation when $3x = \frac{13\pi}{3}$ or $3x = \frac{17\pi}{3}$.

Q: What is the significance of the interval $[0, \pi]$ in solving the equation $2 \cos 3x = 1$?

A: The interval $[0, \pi]$ is significant in solving the equation $2 \cos 3x = 1$ because it restricts the values of $x$ to a specific range. This allows us to find the solutions to the equation in a more manageable way.

Q: How can we use the solutions to the equation $2 \cos 3x = 1$ in real-world applications?

A: The solutions to the equation $2 \cos 3x = 1$ can be used in various real-world applications, such as modeling the motion of a pendulum or the vibration of a spring. The solutions can also be used to find the maximum and minimum values of a function.

Q: What are some common mistakes to avoid when solving the equation $2 \cos 3x = 1$?

A: Some common mistakes to avoid when solving the equation $2 \cos 3x = 1$ include:

• Not isolating the cosine function
• Not using the unit circle and the values of the cosine function at different angles
• Not considering the interval $[0, \pi]$ when finding the solutions
• Not checking the solutions to ensure they satisfy the equation

Q: How can we verify the solutions to the equation $2 \cos 3x = 1$?

A: We can verify the solutions to the equation $2 \cos 3x = 1$ by plugging them back into the original equation and checking if they satisfy the equation. We can also use the unit circle and the values of the cosine function at different angles to verify the solutions.

Q: What are some additional resources for learning more about solving trigonometric equations like $2 \cos 3x = 1$?

A: Some additional resources for learning more about solving trigonometric equations like $2 \cos 3x = 1$ include:

• Textbooks on trigonometry and calculus
• Online resources and tutorials
• Practice problems and exercises
• Real-world applications and examples