Solve Cos ⁡ ( X ) ( Cos ⁡ X + 1 ) = 0 \cos(x)(\cos X + 1) = 0 Cos ( X ) ( Cos X + 1 ) = 0 A. X = Π 3 ± 2 Π N , X = 3 Π 4 ± 2 Π N X = \frac{\pi}{3} \pm 2\pi N, \quad X = \frac{3\pi}{4} \pm 2\pi N X = 3 Π ​ ± 2 Πn , X = 4 3 Π ​ ± 2 Πn B. X = Π 2 ± 2 Π N , X = 3 Π 2 ± 2 Π N X = \frac{\pi}{2} \pm 2\pi N, \quad X = \frac{3\pi}{2} \pm 2\pi N X = 2 Π ​ ± 2 Πn , X = 2 3 Π ​ ± 2 Πn C. $x = \frac{\pi}{2} \pm 2\pi N, \quad X =

$$

Introduction

In this article, we will be solving the trigonometric equation $\cos(x)(\cos x + 1) = 0$. This equation involves the product of two cosine functions, and we need to find the values of $x$ that satisfy this equation. We will use various trigonometric identities and properties to solve this equation and find the possible values of $x$.

Understanding the Equation

The given equation is $\cos(x)(\cos x + 1) = 0$. To solve this equation, we need to find the values of $x$ that make the product of $\cos(x)$ and $(\cos x + 1)$ equal to zero. This means that either $\cos(x)$ or $(\cos x + 1)$ must be equal to zero.

Solving for $\cos(x) = 0$

Let's first consider the case where $\cos(x) = 0$. We know that the cosine function is equal to zero at odd multiples of $\frac{\pi}{2}$. Therefore, we can write:

$$\cos(x) = 0 \Rightarrow x = \frac{(2n + 1)\pi}{2}$$

where $n$ is an integer.

Solving for $(\cos x + 1) = 0$

Now, let's consider the case where $(\cos x + 1) = 0$. We can rewrite this equation as:

$$\cos x = -1$$

We know that the cosine function is equal to $-1$ at odd multiples of $\pi$. Therefore, we can write:

$$\cos x = -1 \Rightarrow x = (2n + 1)\pi$$

where $n$ is an integer.

Combining the Solutions

Now, let's combine the solutions from both cases. We have:

$$x = \frac{(2n + 1)\pi}{2} \quad \text{or} \quad x = (2n + 1)\pi$$

We can rewrite these solutions as:

$$x = \frac{\pi}{2} \pm 2\pi n \quad \text{or} \quad x = \frac{3\pi}{2} \pm 2\pi n$$

where $n$ is an integer.

Conclusion

In this article, we have solved the trigonometric equation $\cos(x)(\cos x + 1) = 0$. We have found that the possible values of $x$ are:

$$x = \frac{\pi}{2} \pm 2\pi n \quad \text{or} \quad x = \frac{3\pi}{2} \pm 2\pi n$$

where $n$ is an integer.

Final Answer

The final answer is:

A. $x = \frac{\pi}{2} \pm 2\pi n, \quad x = \frac{3\pi}{2} \pm 2\pi n$

This is the correct solution to the given equation.

Discussion

The solution to the equation $\cos(x)(\cos x + 1) = 0$ involves finding the values of $x$ that make the product of $\cos(x)$ and $(\cos x + 1)$ equal to zero. We have used various trigonometric identities and properties to solve this equation and find the possible values of $x$. The final answer is:

A. $x = \frac{\pi}{2} \pm 2\pi n, \quad x = \frac{3\pi}{2} \pm 2\pi n$

This solution is valid for all values of $n$, where $n$ is an integer.

Additional Information

The equation $\cos(x)(\cos x + 1) = 0$ is a classic example of a trigonometric equation that involves the product of two cosine functions. The solution to this equation involves finding the values of $x$ that make the product of $\cos(x)$ and $(\cos x + 1)$ equal to zero. We have used various trigonometric identities and properties to solve this equation and find the possible values of $x$. The final answer is:

A. $x = \frac{\pi}{2} \pm 2\pi n, \quad x = \frac{3\pi}{2} \pm 2\pi n$

This solution is valid for all values of $n$, where $n$ is an integer.

References

• [1] "Trigonometry" by Michael Corral
• [2] "Calculus" by Michael Spivak
• [3] "Trigonometric Equations" by Paul Dawkins

Note: The references provided are for general information and are not directly related to the solution of the equation $\cos(x)(\cos x + 1) = 0$.

$$

Introduction

In our previous article, we solved the trigonometric equation $\cos(x)(\cos x + 1) = 0$ and found that the possible values of $x$ are:

$$x = \frac{\pi}{2} \pm 2\pi n \quad \text{or} \quad x = \frac{3\pi}{2} \pm 2\pi n$$

where $n$ is an integer. In this article, we will answer some frequently asked questions related to this equation.

Q1: What is the significance of the equation $\cos(x)(\cos x + 1) = 0$?

A1: The equation $\cos(x)(\cos x + 1) = 0$ is a classic example of a trigonometric equation that involves the product of two cosine functions. It is used to model various real-world phenomena, such as the motion of a pendulum or the vibration of a spring.

Q2: How do I solve the equation $\cos(x)(\cos x + 1) = 0$?

A2: To solve the equation $\cos(x)(\cos x + 1) = 0$, you need to find the values of $x$ that make the product of $\cos(x)$ and $(\cos x + 1)$ equal to zero. This involves using various trigonometric identities and properties, such as the Pythagorean identity and the unit circle.

Q3: What are the possible values of $x$ that satisfy the equation $\cos(x)(\cos x + 1) = 0$?

A3: The possible values of $x$ that satisfy the equation $\cos(x)(\cos x + 1) = 0$ are:

$$x = \frac{\pi}{2} \pm 2\pi n \quad \text{or} \quad x = \frac{3\pi}{2} \pm 2\pi n$$

where $n$ is an integer.

Q4: How do I graph the equation $\cos(x)(\cos x + 1) = 0$?

A4: To graph the equation $\cos(x)(\cos x + 1) = 0$, you need to plot the graphs of $\cos(x)$ and $(\cos x + 1)$ on the same coordinate plane. The points of intersection of these two graphs represent the solutions to the equation.

Q5: What are some real-world applications of the equation $\cos(x)(\cos x + 1) = 0$?

A5: The equation $\cos(x)(\cos x + 1) = 0$ has various real-world applications, such as:

• Modeling the motion of a pendulum or a spring
• Analyzing the vibration of a structure or a machine
• Studying the behavior of a physical system under different conditions

Q6: How do I use the equation $\cos(x)(\cos x + 1) = 0$ in engineering or physics?

A6: To use the equation $\cos(x)(\cos x + 1) = 0$ in engineering or physics, you need to apply it to a specific problem or scenario. For example, you might use it to model the motion of a pendulum or a spring, or to analyze the vibration of a structure or a machine.

Q7: What are some common mistakes to avoid when solving the equation $\cos(x)(\cos x + 1) = 0$?

A7: Some common mistakes to avoid when solving the equation $\cos(x)(\cos x + 1) = 0$ include:

• Failing to use the correct trigonometric identities or properties
• Not considering all possible solutions or cases
• Making errors in graphing or plotting the equation

Q8: How do I verify the solutions to the equation $\cos(x)(\cos x + 1) = 0$?

A8: To verify the solutions to the equation $\cos(x)(\cos x + 1) = 0$, you need to substitute each solution into the original equation and check if it is true. You can also use graphing or plotting tools to visualize the solutions and verify their accuracy.

Q9: What are some advanced topics related to the equation $\cos(x)(\cos x + 1) = 0$?

A9: Some advanced topics related to the equation $\cos(x)(\cos x + 1) = 0$ include:

• Using the equation to model more complex systems or phenomena
• Applying the equation to different fields or disciplines, such as physics, engineering, or mathematics
• Developing new techniques or methods for solving the equation or related problems

Q10: How do I learn more about the equation $\cos(x)(\cos x + 1) = 0$ and related topics?

A10: To learn more about the equation $\cos(x)(\cos x + 1) = 0$ and related topics, you can:

• Consult textbooks or online resources, such as Khan Academy or Wolfram Alpha
• Take online courses or attend workshops or conferences related to trigonometry or mathematics
• Practice solving problems and exercises related to the equation and its applications.