Find All Solutions To 2 Cos ⁡ ( Θ ) ≤ − 1 2 \cos (\theta) \leq -1 2 Cos ( Θ ) ≤ − 1 On The Interval 0 ≤ Θ \textless 2 Π 0 \leq \theta \ \textless \ 2\pi 0 ≤ Θ \textless 2 Π . Be Sure To Write Your Answer In Interval Notation.

Introduction

In this article, we will explore the solutions to the inequality $2 \cos (\theta) \leq -1$ on the interval $0 \leq \theta \ \textless \ 2\pi$. This involves understanding the behavior of the cosine function and its relationship with the given inequality. We will use mathematical techniques to find the solutions and express them in interval notation.

Understanding the Cosine Function

The cosine function, denoted as $\cos (\theta)$, is a fundamental trigonometric function that describes the ratio of the adjacent side to the hypotenuse in a right-angled triangle. The cosine function has a periodic nature, with a period of $2\pi$, meaning that it repeats its values every $2\pi$ radians. The cosine function is also an even function, which means that $\cos (-\theta) = \cos (\theta)$.

Graph of the Cosine Function

The graph of the cosine function is a sinusoidal curve that oscillates between the values of $-1$ and $1$. The graph has a maximum value of $1$ at $\theta = 0$ and a minimum value of $-1$ at $\theta = \pi$. The graph also has a period of $2\pi$, with the same shape and amplitude repeating every $2\pi$ radians.

Solving the Inequality

To solve the inequality $2 \cos (\theta) \leq -1$, we need to isolate the cosine function. Dividing both sides of the inequality by $2$, we get:

$$\cos (\theta) \leq -\frac{1}{2}$$

This inequality is satisfied when the cosine function is less than or equal to $-\frac{1}{2}$. We can use the graph of the cosine function to visualize the solution.

Finding the Solutions

Using the graph of the cosine function, we can see that the cosine function is less than or equal to $-\frac{1}{2}$ in the intervals $[\frac{2\pi}{3}, \pi]$ and $[\frac{4\pi}{3}, \frac{5\pi}{3}]$. These intervals correspond to the values of $\theta$ that satisfy the inequality.

Expressing the Solutions in Interval Notation

The solutions to the inequality $2 \cos (\theta) \leq -1$ on the interval $0 \leq \theta \ \textless \ 2\pi$ can be expressed in interval notation as:

$$\left[\frac{2\pi}{3}, \pi\right] \cup \left[\frac{4\pi}{3}, \frac{5\pi}{3}\right]$$

This expression represents the two intervals where the cosine function is less than or equal to $-\frac{1}{2}$.

Conclusion

In this article, we have explored the solutions to the inequality $2 \cos (\theta) \leq -1$ on the interval $0 \leq \theta \ \textless \ 2\pi$. We have used mathematical techniques to understand the behavior of the cosine function and its relationship with the given inequality. We have also expressed the solutions in interval notation, providing a clear and concise representation of the solution set.

Final Answer

The final answer is $\boxed{\left[\frac{2\pi}{3}, \pi\right] \cup \left[\frac{4\pi}{3}, \frac{5\pi}{3}\right]}$.

Introduction

In our previous article, we explored the solutions to the inequality $2 \cos (\theta) \leq -1$ on the interval $0 \leq \theta \ \textless \ 2\pi$. We used mathematical techniques to understand the behavior of the cosine function and its relationship with the given inequality. In this article, we will provide a Q&A section to help clarify any doubts or questions that readers may have.

Q: What is the significance of the interval $0 \leq \theta \ \textless \ 2\pi$ in this problem?

A: The interval $0 \leq \theta \ \textless \ 2\pi$ represents the range of values that $\theta$ can take. In this problem, we are interested in finding the solutions to the inequality $2 \cos (\theta) \leq -1$ within this interval.

Q: How do we know that the cosine function is periodic with a period of $2\pi$?

A: The cosine function is periodic with a period of $2\pi$ because it repeats its values every $2\pi$ radians. This means that the graph of the cosine function will have the same shape and amplitude repeating every $2\pi$ radians.

Q: What is the relationship between the cosine function and the given inequality?

A: The cosine function is related to the given inequality through the equation $2 \cos (\theta) \leq -1$. This inequality is satisfied when the cosine function is less than or equal to $-\frac{1}{2}$.

Q: How do we find the solutions to the inequality $2 \cos (\theta) \leq -1$?

A: To find the solutions to the inequality $2 \cos (\theta) \leq -1$, we need to isolate the cosine function. Dividing both sides of the inequality by $2$, we get $\cos (\theta) \leq -\frac{1}{2}$. We can then use the graph of the cosine function to visualize the solution.

Q: What are the solutions to the inequality $2 \cos (\theta) \leq -1$ on the interval $0 \leq \theta \ \textless \ 2\pi$?

A: The solutions to the inequality $2 \cos (\theta) \leq -1$ on the interval $0 \leq \theta \ \textless \ 2\pi$ can be expressed in interval notation as $\left[\frac{2\pi}{3}, \pi\right] \cup \left[\frac{4\pi}{3}, \frac{5\pi}{3}\right]$.

Q: Why is it important to express the solutions in interval notation?

A: Expressing the solutions in interval notation provides a clear and concise representation of the solution set. It allows us to easily identify the intervals where the cosine function is less than or equal to $-\frac{1}{2}$.

Q: Can you provide more examples of how to solve inequalities involving the cosine function?

A: Yes, we can provide more examples of how to solve inequalities involving the cosine function. For example, consider the inequality $\cos (\theta) \geq \frac{1}{2}$. We can use the same techniques as before to find the solutions to this inequality.

Q: How do we know that the solutions to the inequality $2 \cos (\theta) \leq -1$ are unique?

A: The solutions to the inequality $2 \cos (\theta) \leq -1$ are unique because the cosine function is a continuous and monotonic function on the interval $0 \leq \theta \ \textless \ 2\pi$. This means that the graph of the cosine function will not intersect with itself within this interval.

Q: Can you provide a visual representation of the solutions to the inequality $2 \cos (\theta) \leq -1$?

A: Yes, we can provide a visual representation of the solutions to the inequality $2 \cos (\theta) \leq -1$. We can use a graphing calculator or software to plot the graph of the cosine function and identify the intervals where the function is less than or equal to $-\frac{1}{2}$.

Conclusion

In this Q&A article, we have provided answers to common questions that readers may have about the problem of finding the solutions to the inequality $2 \cos (\theta) \leq -1$ on the interval $0 \leq \theta \ \textless \ 2\pi$. We hope that this article has helped to clarify any doubts or questions that readers may have.