Evaluate Arccsc ⁡ 2 \operatorname{arccsc} \sqrt{2} Arccsc 2 ​ .A. { Π 6 , 5 Π 6 } \left\{\frac{\pi}{6}, \frac{5 \pi}{6}\right\} { 6 Π ​ , 6 5 Π ​ } B. { Π 3 ± 2 Π N , 5 Π 3 ± 2 Π N } \left\{\frac{\pi}{3} \pm 2 \pi N, \frac{5 \pi}{3} \pm 2 \pi N\right\} { 3 Π ​ ± 2 Πn , 3 5 Π ​ ± 2 Πn } C. Π 3 \frac{\pi}{3} 3 Π ​ D. $\left{\frac{\pi}{4} \pm

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Introduction

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In mathematics, the inverse cosecant function, denoted as $\operatorname{arccsc}$, is the inverse of the cosecant function. It returns the angle whose cosecant is a given value. In this article, we will evaluate the inverse cosecant of the square root of 2, denoted as $\operatorname{arccsc} \sqrt{2}$.

Understanding the Inverse Cosecant Function

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The inverse cosecant function is defined as the angle whose cosecant is a given value. In other words, if $\csc(x) = y$, then $\operatorname{arccsc}(y) = x$. The range of the inverse cosecant function is $[0, \pi]$.

Evaluating the Inverse Cosecant of the Square Root of 2

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To evaluate $\operatorname{arccsc} \sqrt{2}$, we need to find the angle whose cosecant is $\sqrt{2}$. We can start by finding the cosecant of some common angles and see if we can find a match.

Common Angles and Their Cosecant Values

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Angle	Cosecant Value
$\frac{\pi}{6}$	$\sqrt{3}$
$\frac{\pi}{4}$	$\sqrt{2}$
$\frac{\pi}{3}$	$\frac{2}{\sqrt{3}}$
$\frac{\pi}{2}$	$\infty$

From the table above, we can see that the cosecant of $\frac{\pi}{4}$ is $\sqrt{2}$. Therefore, $\operatorname{arccsc} \sqrt{2} = \frac{\pi}{4}$.

Considering the Periodicity of the Inverse Cosecant Function

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The inverse cosecant function has a periodicity of $2\pi$, which means that the function repeats itself every $2\pi$ radians. Therefore, we can add or subtract multiples of $2\pi$ to the angle $\frac{\pi}{4}$ to get other possible values of $\operatorname{arccsc} \sqrt{2}$.

Possible Values of $\operatorname{arccsc} \sqrt{2}$

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Using the periodicity of the inverse cosecant function, we can find the following possible values of $\operatorname{arccsc} \sqrt{2}$:

• $\frac{\pi}{4}$
• $\frac{\pi}{4} + 2\pi = \frac{9\pi}{4}$
• $\frac{\pi}{4} - 2\pi = -\frac{7\pi}{4}$

Conclusion

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In conclusion, the possible values of $\operatorname{arccsc} \sqrt{2}$ are $\frac{\pi}{4}$, $\frac{9\pi}{4}$, and $-\frac{7\pi}{4}$. Therefore, the correct answer is:

• $\boxed{\left\{\frac{\pi}{4} \pm 2 \pi n\right\}}$

Note that the answer is in the form of a set, which represents the possible values of $\operatorname{arccsc} \sqrt{2}$.

Final Answer

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The answer is $\boxed{\left\{\frac{\pi}{4} \pm 2 \pi n\right\}}$.

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Introduction

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In our previous article, we evaluated the inverse cosecant of the square root of 2, denoted as $\operatorname{arccsc} \sqrt{2}$. We found that the possible values of $\operatorname{arccsc} \sqrt{2}$ are $\frac{\pi}{4}$, $\frac{9\pi}{4}$, and $-\frac{7\pi}{4}$. In this article, we will answer some frequently asked questions related to the inverse cosecant function and its evaluation.

Q&A

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Q: What is the range of the inverse cosecant function?

A: The range of the inverse cosecant function is $[0, \pi]$.

Q: How do I evaluate the inverse cosecant of a given value?

A: To evaluate the inverse cosecant of a given value, you need to find the angle whose cosecant is the given value. You can use a calculator or a trigonometric table to find the angle.

Q: What is the periodicity of the inverse cosecant function?

A: The inverse cosecant function has a periodicity of $2\pi$, which means that the function repeats itself every $2\pi$ radians.

Q: Can I use the inverse cosecant function to find the angle whose cosecant is a negative value?

A: Yes, you can use the inverse cosecant function to find the angle whose cosecant is a negative value. However, you need to be careful when using the function, as the range of the inverse cosecant function is $[0, \pi]$.

Q: How do I find the inverse cosecant of a complex number?

A: To find the inverse cosecant of a complex number, you need to use the complex inverse cosecant function, denoted as $\operatorname{arccsc}(z)$. This function returns the complex angle whose cosecant is the given complex number.

Q: Can I use the inverse cosecant function to solve trigonometric equations?

A: Yes, you can use the inverse cosecant function to solve trigonometric equations. However, you need to be careful when using the function, as the range of the inverse cosecant function is $[0, \pi]$.

Example Problems

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Problem 1: Evaluate $\operatorname{arccsc} \sqrt{3}$.

A: To evaluate $\operatorname{arccsc} \sqrt{3}$, we need to find the angle whose cosecant is $\sqrt{3}$. We can use a calculator or a trigonometric table to find the angle. The angle whose cosecant is $\sqrt{3}$ is $\frac{\pi}{3}$.

Problem 2: Evaluate $\operatorname{arccsc} (-\sqrt{2})$.

A: To evaluate $\operatorname{arccsc} (-\sqrt{2})$, we need to find the angle whose cosecant is $-\sqrt{2}$. We can use a calculator or a trigonometric table to find the angle. The angle whose cosecant is $-\sqrt{2}$ is $-\frac{\pi}{4}$.

Conclusion

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In conclusion, the inverse cosecant function is a powerful tool for evaluating trigonometric expressions. By understanding the range periodicity of the function, you can use it to solve a wide range of problems. We hope that this article has been helpful in answering your questions about the inverse cosecant function.

Final Answer

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The final answer is $\boxed{\left\{\frac{\pi}{4} \pm 2 \pi n\right\}}$.