Solve Cos ⁡ 2 ( X ) − Cos ⁡ ( X ) = 0 \cos^2(x) - \cos(x) = 0 Cos 2 ( X ) − Cos ( X ) = 0 For X X X , Where 0 ≤ X ≤ 2 Π 0 \leq X \leq 2\pi 0 ≤ X ≤ 2 Π . (Select All That Apply.)A. 0 0 0 B. Π 2 \frac{\pi}{2} 2 Π ​ C. Π \pi Π D. 3 Π 2 \frac{3\pi}{2} 2 3 Π ​ E. 2 Π 2\pi 2 Π

Introduction

In this article, we will delve into solving the trigonometric equation $\cos^2(x) - \cos(x) = 0$ for $x$, where $0 \leq x \leq 2\pi$. This equation involves a quadratic expression in terms of $\cos(x)$, and we will employ various techniques to find the solutions.

Understanding the Equation

The given equation is $\cos^2(x) - \cos(x) = 0$. We can rewrite this equation as $\cos(x)(\cos(x) - 1) = 0$. This equation is a product of two factors, and we can use the zero-product property to find the solutions.

Applying the Zero-Product Property

The zero-product property states that if $ab = 0$, then either $a = 0$ or $b = 0$. Applying this property to our equation, we get:

$$\cos(x) = 0 \quad \text{or} \quad \cos(x) - 1 = 0$$

Solving the First Equation

The first equation is $\cos(x) = 0$. We know that the cosine function has a period of $2\pi$, and it is equal to zero at $x = \frac{\pi}{2}$ and $x = \frac{3\pi}{2}$ in the interval $[0, 2\pi]$. Therefore, the solutions to this equation are:

$$x = \frac{\pi}{2}, \frac{3\pi}{2}$$

Solving the Second Equation

The second equation is $\cos(x) - 1 = 0$. We can rewrite this equation as $\cos(x) = 1$. The cosine function is equal to one at $x = 0$ and $x = 2\pi$ in the interval $[0, 2\pi]$. Therefore, the solution to this equation is:

$$x = 0, 2\pi$$

Combining the Solutions

We have found the solutions to both equations, and we can combine them to get the final solutions to the original equation. Therefore, the solutions to the equation $\cos^2(x) - \cos(x) = 0$ for $x$, where $0 \leq x \leq 2\pi$ are:

$$x = 0, \frac{\pi}{2}, 2\pi, \frac{3\pi}{2}$$

Conclusion

In this article, we have solved the trigonometric equation $\cos^2(x) - \cos(x) = 0$ for $x$, where $0 \leq x \leq 2\pi$. We have employed the zero-product property and used the properties of the cosine function to find the solutions. The final solutions are $x = 0, \frac{\pi}{2}, 2\pi, \frac{3\pi}{2}$.

Answer Key

The correct answers are:

• A. $0$
• B. $\frac{\pi}{2}$
• C. $\pi$
• D. $\frac{3\pi}{2}$
• E. $2\pi$

Q: What is the zero-product property?

A: The zero-product property states that if $ab = 0$, then either $a = 0$ or $b = 0$. This property is used to find the solutions to the equation $\cos^2(x) - \cos(x) = 0$.

Q: How do I apply the zero-product property to the equation?

A: To apply the zero-product property, we can rewrite the equation as $\cos(x)(\cos(x) - 1) = 0$. Then, we can use the property to find the solutions: $\cos(x) = 0$ or $\cos(x) - 1 = 0$.

Q: What are the solutions to the equation $\cos(x) = 0$?

A: The solutions to the equation $\cos(x) = 0$ are $x = \frac{\pi}{2}$ and $x = \frac{3\pi}{2}$ in the interval $[0, 2\pi]$.

Q: What are the solutions to the equation $\cos(x) - 1 = 0$?

A: The solution to the equation $\cos(x) - 1 = 0$ is $x = 0$ and $x = 2\pi$ in the interval $[0, 2\pi]$.

Q: How do I combine the solutions to get the final solutions to the equation?

A: To combine the solutions, we can simply list all the solutions we found: $x = 0, \frac{\pi}{2}, 2\pi, \frac{3\pi}{2}$.

Q: What is the final answer to the equation $\cos^2(x) - \cos(x) = 0$?

A: The final answer to the equation $\cos^2(x) - \cos(x) = 0$ is $x = 0, \frac{\pi}{2}, 2\pi, \frac{3\pi}{2}$.

Q: Why is $\pi$ not a solution to the equation?

A: $\pi$ is not a solution to the equation because it does not satisfy the equation $\cos^2(x) - \cos(x) = 0$. The cosine function is equal to one at $x = \pi$, but this does not make $\pi$ a solution to the equation.

Q: Can I use other methods to solve the equation?

A: Yes, you can use other methods to solve the equation, such as using the quadratic formula or factoring. However, the zero-product property is a simple and effective method to solve this type of equation.

Q: What is the importance of solving trigonometric equations?

A: Solving trigonometric equations is important because it helps us understand the behavior of trigonometric functions and how they relate to each other. This knowledge is essential in many areas of mathematics and science, such as physics, engineering, and computer science.

Q: Can I apply the zero-product property to other types of equations?

A: Yes, the zero-product property can be applied to other types of equations, such as polynomial equations. However, the specific method of applying the property may vary depending on the type of equation.