For Which Value Of $\theta$ Is $\csc (\theta)$ Undefined?A. 0 B. \$\frac{\pi}{2}$[/tex\] C. $\frac{3 \pi}{2}$ D. $\frac{7 \pi}{2}$

Understanding the Domain of the Cosecant Function

The cosecant function, denoted as $\csc (\theta)$, is the reciprocal of the sine function. It is defined as $\csc (\theta) = \frac{1}{\sin (\theta)}$. In order to understand for which value of $\theta$ the cosecant function is undefined, we need to consider the domain of the sine function.

The Domain of the Sine Function

The sine function is defined for all real numbers, but it has certain values that make it undefined. The sine function is undefined when its denominator is equal to zero. In other words, $\sin (\theta)$ is undefined when $\sin (\theta) = 0$. This occurs when $\theta$ is an integer multiple of $\pi$, i.e., $\theta = k \pi$, where $k$ is an integer.

The Reciprocal of the Sine Function

Since the cosecant function is the reciprocal of the sine function, it is also undefined when the sine function is equal to zero. Therefore, the cosecant function is undefined when $\sin (\theta) = 0$. This occurs when $\theta$ is an integer multiple of $\pi$, i.e., $\theta = k \pi$, where $k$ is an integer.

Finding the Value of $\theta$

We are given four options for the value of $\theta$ for which the cosecant function is undefined. Let's evaluate each option:

• Option A: $\theta = 0$. This is an integer multiple of $\pi$, so the sine function is equal to zero, and the cosecant function is undefined.
• Option B: $\theta = \frac{\pi}{2}$. This is not an integer multiple of $\pi$, so the sine function is not equal to zero, and the cosecant function is defined.
• Option C: $\theta = \frac{3 \pi}{2}$. This is an integer multiple of $\pi$, so the sine function is equal to zero, and the cosecant function is undefined.
• Option D: $\theta = \frac{7 \pi}{2}$. This is an integer multiple of $\pi$, so the sine function is equal to zero, and the cosecant function is undefined.

Conclusion

Based on our analysis, we can conclude that the cosecant function is undefined when $\theta$ is an integer multiple of $\pi$. Therefore, the correct answer is:

• Option A: $\theta = 0$
• Option C: $\theta = \frac{3 \pi}{2}$
• Option D: $\theta = \frac{7 \pi}{2}$

However, since the question asks for a single value of $\theta$, we can choose any one of the above options as the correct answer. In this case, we will choose option A: $\theta = 0$.

Final Answer

The final answer is $\boxed{0}$.
Q&A: Understanding the Domain of the Cosecant Function

In our previous article, we discussed the domain of the cosecant function and how it is related to the sine function. We also analyzed the given options and concluded that the cosecant function is undefined when $\theta$ is an integer multiple of $\pi$. In this article, we will provide a Q&A section to help clarify any doubts and provide further understanding of the topic.

Q: What is the cosecant function?

A: The cosecant function, denoted as $\csc (\theta)$, is the reciprocal of the sine function. It is defined as $\csc (\theta) = \frac{1}{\sin (\theta)}$.

Q: When is the cosecant function undefined?

A: The cosecant function is undefined when the sine function is equal to zero. This occurs when $\theta$ is an integer multiple of $\pi$, i.e., $\theta = k \pi$, where $k$ is an integer.

Q: Why is the cosecant function undefined when the sine function is equal to zero?

A: The cosecant function is the reciprocal of the sine function. When the sine function is equal to zero, the cosecant function would have a denominator of zero, which is undefined.

Q: What are the values of $\theta$ for which the cosecant function is undefined?

A: The cosecant function is undefined when $\theta$ is an integer multiple of $\pi$. This includes values such as $\theta = 0, \frac{\pi}{2}, \pi, \frac{3 \pi}{2}, ...$.

Q: Can you provide an example of how to find the value of $\theta$ for which the cosecant function is undefined?

A: Let's consider the option $\theta = \frac{\pi}{2}$. We know that the sine function is equal to one when $\theta = \frac{\pi}{2}$. Therefore, the cosecant function would be undefined at this value of $\theta$.

Q: How can we use the cosecant function in real-world applications?

A: The cosecant function has applications in various fields such as trigonometry, calculus, and physics. It can be used to model periodic phenomena, such as the motion of a pendulum or the vibration of a spring.

Q: What are some common mistakes to avoid when working with the cosecant function?

A: Some common mistakes to avoid when working with the cosecant function include:

• Assuming that the cosecant function is defined for all values of $\theta$.
• Failing to check if the sine function is equal to zero before evaluating the cosecant function.
• Not considering the domain of the cosecant function when solving equations or inequalities.

Conclusion

In this Q&A article, we have provided further clarification on the domain of the cosecant function and its relationship to the sine function. We have also addressed common questions and provided examples to help illustrate the concepts. By understanding the domain of the cosecant function, we can better appreciate its applications and avoid common mistakes.

Final Answer

The final answer is $\boxed{0}$.